Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations
1969 The production function and the imputation of the economic value of irrigation water Alan Pilling Kleinman Iowa State University
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KLEINMM, Alan Pilling, 1936- THE PRODUCTION FUNCTION AND THE IMPUTATION OF THE ECONOMIC VALUE OF IRRIGATION WATER.
Iowa State University, Ph.D., 1969 Economics, agricultural
University Microfilms, Inc., Ann Arbor, Michigan
THIS DISSERTATION HAS' BEEN MICROFILMED EXACTLY AS RECEIVED THE PRODUCTION FUNCTION AND TOE IMPUTATION OF THE ECONOMIC
VALUE OF IRRIGATION WATER
by
Alan Pilling Kleinman
A Dissertation Submitted to the
Graduate Faculty in Partial Fulfillment of
The Requirements for the Degree of
DOCTOR OF PHILOSOPHY
Major Subject: Agricultural Economics
Approved:
Signature was redacted for privacy.
Signature was redacted for privacy.
Head of Major Department
Signature was redacted for privacy.
Iowa State University Ames, Iowa
1969 PLEASE NOTE:
Not original copy. Some pages have indistinct print. Filmed as received.
UNIVERSITY MICROFILMS il
TABLE OF CONTENTS Page
INTRODUCTION AND STUDY JUSTIFICATION 1
Area and scope of study 1 The problem 3 Objectives of study 7 Analytical techniques and procedures 8 Plan of study 8
THE THEORY OF THE PRODUCTION FUNCTION 10
Production function defined 11a Type of production functions 12 Physical production functions 14 Cobb-Douglas function 16 Spillman function 17 Quadratic forms 17 Derivations of production functions 18 Marginal rate of substitution 18 Isoquants 18 Isoclines 19 Profit maximization 20 Appropriate functions 21 Selection of the form of the function 21
WATER-PLANT RELATIONSHIPS 23
Factors affecting moisture availability 29 Soil factors 32 Plant growth stage and growth response criterion 32 Climatic factors 34 Plant response to water 34 Moisture movement through the soil at irrigation 39 Plant growth response 42 The "equal facility" theory 44 The "inverse function" theory 48 Integrated moisture stress 52 Wadleigh's method 54 Tay1or's method 55 Generalized relationship 56 Irrigation scheduling model 57 Summary 62 iii
TABLE OF CONTENTS (Continued) Page
THE PRODUCTION FUNCTION FOR CROPS USING IRRIGATION WATER 65
Towards a water production function 67 Assumptions 68 Nature of the model 69 Time as a variable 72 Profit maximizing solution 76 The dynamic nature of crop response 79 Summary 81
AN EMPIRICAL PRODUCTION FUNCTION FOR CORN USING IRRIGATION WATER AND NITROGEN FERTILIZER 84
Framework of the corn experiment 85 Production function estimation 87 Moisture tension and nitrogen as variables 93 Inches of irrigation water applied and nitrogen as variables 96 Yield predictions 101 Summary 118
ALTERNATIVE METHODS OF VALUING IRRIGATION WATER 119
Budgeting or residual imputation 121 A farm production function 125 Linear programming 12 8 Comparing linear programming with budgeting 130 Simplex tableau for a linear programming problem 131 Economic interpretation of simplex procedures 137 Other applications of linear programming 138 Estimating the demand for irrigation water 139 Classical production function and marginal analysis 143 Euler's theorem 146 Imputing the value of irrigation water using empirical production functions 149 Summary 156
SUMMARY 159
SELECTED BIBLIOGRAPHY 165
ACKNOWLEDGMENTS 169
APPENDIX 170 Irrigation scheduling model results using criterion I 172 Irrigation scheduling model results using criterion II 188 Irrigation scheduling model results using criterion III 204 Irrigation scheduling model results using criterion IV 220 iv
LIST OF TABLES Table Page
1 Water value, dollars per acre foot 2
2 Hypothetical crop production and irrigation cost data 78
3 Plot grain yields on pounds per acre basis adjusted to 15.5 percent moisture 88
4 Plot forage yields on pounds per acre basis oven dry weights 89
5 Conversion table from coded X values to actual X values for the production function based upon moisture stress 91
5 Relationship between the soil moisture stress and the seasonal inches of irrigation water applied 91
7 Grain ANOVA using moisture tension 93
8 Forage ANOVA using moisture tension 94
9 Grain ANOVA with error pooled using moisture tension 94
10 Forage ANOVA with error pooled using moisture tension 95
11 Coefficients of grain production function using moisture tension 95
12 Coefficients of forage production function using moisture tension 96
13 Conversion table from coded X values to actual X values for the production function based upon inches of water applied 97
14 Grain ANOVA using inches of water applied 98
15 Forage ANOVA using inches of water applied 98
16 Grain ANOVA with error pooled using inches of water applied 99
17 Forage ANOVA with error pooled using inches of water applied 99 V
LIST OF TABLES Table (Continued) Page
18 Coefficients of grain production function using inches of water applied 100
19 Coefficients of forage production function using inches of water applied 100
2 0 Predicted yields of corn grain in pounds per acre for se lected nitrogen levels and soil moisture tensions 102
21 Predicted yields of corn forage in pounds per acre for selected nitrogen levels and soil moisture tensions 104
22 Predicted yields of corn grain in bushels per acre for selected nitrogen levels and soil moisture tensions 106
23 Predicted yields of corn forage in tons per acre for selected nitrogen levels and soil moisture tensions 108
24 Predicted yields of corn grain in pounds per acre for selected nitrogen levels and irrigation water applications 110
25 Predicted yields of corn forage in pounds per acre for selected nitrogen levels and irrigation water applications 112
26 Predicted yields of corn grain in bushels per acre for selected nitrogen levels and irrigation water applications
27 Predicted yields of corn forage in tons per acre for selected nitrogen levels and irrigation water applications 116
28 Data for programming 133
29 Simplex tableau 135
30 Summary of solutions to variable price programming for water delivered to the farm 1^2
31 Quantities of water to use for profit maximization for various water prices and nitrogen applications with a grain price of 1.78 cents per pound 153
32 Profit maximizing quantities of water and nitrogen for corn grain with five prices of water 155 vi
LIST OF FIGURES Figure Page
1 The volumes of solids, water, and air in a silt loam soil at different moisture levels 25
2 The relationship between thickness of water films and the tension with which the water is held at the liquid air interface 26
3 The general relationship between soil moisture charac teristics and soil texture 27
4 Comparative rates of irrigation water movement into a sandy loam and a clay loam soil 28
5 Variations in how relative growth relates to available moisture depletion for sandy, loam, and clay soils 30
6 Moisture retention on soil core with natural structure 36
7 Growth of bean plants as influenced by the soil moisture stress 37
8 Elongation of cotton leaves in relation to irrigation 38
9 Pattern of flow of water through a hypothetical homogeneous soil profile 40
10 Effect of soil upon the type of water movement in a profile 41
11 Rate of growth in relation to the moisture content of the soil 43
12 Hypothesized soil moisture-plant growth relationship under the "equal facility" theory 45
13 Relation of moisture stress to moisture percentage at different levels of soil salinity 47
14 Curve showing general relationship between the content of available water in the soil and the yield of crops 50
15 A production function for one irrigation cycle (hypothetical) 71
16 Discrete water application over time 73
17 Nature of crop response to successive irrigations 74 vii
LIST OF FIGURES Figure Page
18 Costs and returns for hypothetical crop under irrigation 77
19 Hypothetical yield alternatives with varying irrigation strategies 80
20 Experimental design showing combinations of factors which comprise the treatments 86
21 Graph showing relationship between moisture stress and water applied 92
22 Combining land and water with three crops to maximize income 126
23 Normative demand curve for water delivered at the farm 141
24 Static water demand curves with constant nitrogen and grain prices with three levels of nitrogen application 154 1
INTRODUCTION AND STUDY JUSTIFICATION
In various fields of endeavor, individuals make specific assumptions
and then deduce sets of rules that must be followed to obtain certain
results. One such field of endeavor is economics. In this study the
specific rule used is the law of equimarginal returns; from this rule
can be deduced an optimal* economic allocation of a given resource. The
resource under consideration here is irrigation water applied to crops, which generally has a continuous and predictable supply and is
periodically replenished.
Area and scope of study
The allocation of a natural resource such as water is basically a normative problem. In order to make proper decisions regarding allo cation, a value must be placed on the resource, in this case on water, even though economists are not in complete agreement as to the proper techniques to be used in its valuation. Unfortunately, due to legal institutions that now surround the use of water, it is not generally subject to price allocation. This leaves the determination of value to any agency or group attempting to alter present use or allocation.
In agriculture, the present users of water must demonstrate that water is being used in an efficient manner and that irrigation is economically feasible in relation to alternative uses so that, in competition with other uses, the present allocation will be justified.
•k "Optimal" can be defined in the most general terms as the situation or state of affairs which yields the best or most favorable degree of some desirable property. 2
In the United States, water supply has become sufficiently limited
against demand that only rarely can water be used for one particular
purpose without adversely affecting its use for some other purpose-
Estimates of the value of water in alternative uses have been given by
E. F. Renshaw in a 1958 article.^ The average and maximum values observed
in various uses are given in Table 1. For the most efficient allocation,
the units of resources must be employed so that the value of the marginal
product* is equal in all applications.
Table 1. Water value, dollars per acre foot
Use Average Maximum
Domestic 100.19 235.66
Industrial 40.73 163.3 5
Irrigation 1.67 27.04
Power .71 5.90
Waste disposal .63 2.56
Navigation .05 1.17
Fisheries .025 1.06
In some sense the available supplies of land and water are limited in the United States. Assuming that the effective quantities of these
^E. F. Renshaw. Value of an acre foot of water. Journal of American Water Works Association 50: 303-309. 1958.
The value of the marginal product is defined as the price of the output multiplied by the marginal physical product of the factor, the marginal physical product being the first partial derivative of the production function. 3
two resources can be increased only under limited resource substitution,
the physical combination of the two in irrigation projects then becomes
important. Certainly, land and water are themselves substitutes one
for another, and, as such, demand decision rules to insure that the
combination is optimal. This combination implies knowledge of the
response of crops to varying amounts of land and water.
Before it is possible to employ the use of equimarginal principles,
one must have an estimate of the input-output function underlying the
production process. Without some such idea, it is impossible to make
correct allocation decisions. Many instances of economic investigations
appear to proceed without such information, but upon careful examination
it is usually discovered that some sort of crude assumptions have been
made about the production process, even though these assumptions may
be quite naive. Hence, any attempts to use marginal principles in
economic analysis must include some reference to the underlying production
function.
The areas of concern in this study revolve mainly around the water production function and the imputation of the economic value of
irrigation water.
The problem
Whenever any production process involves the passage of time for completion of the production cycle the entrepreneur or planner subjects himself to the uncertainty that conditions will change before completion of the process. Prices are subject to fluctuation and are uncontrollable 4
by the individual under competitive assumptions. Conditions and factors
surrounding and affecting the production process may also change as
might knowledge about the production function itself. There may even exist great uncertainty about the actual form of the underlying
relationships between water input and crop output.
The time involved in agricultural production does preclude perfect knowledge of the future; therefore, decision-making must take place in an environment of uncertainty. Uncertainty arises because the entreprenuer must formulate an image of the future in his mind but has no quantitative manner by which these predictions can be verified. In short, uncertainty refers to future events where the parameters of the probability distribution cannot be determined empirically.
In this study, the author chooses to ignore the uncertainties of price and factors affecting the production process and assume that perfect foresight exists. But surely, by proper experimentation and research the parameters of the input-output relationships can be determined. To take a step in the direction of conceptualizing and measuring these relationships is the major purpose of the present thesis.
The economics of irrigation development concerns itself with the allocation of scarce resources. The problem is basically one of specifying that combination of resources for production which will make the greatest possible contribution to the welfare of the total economy and society in general. Some relevant questions to be answered are: How should development take place? What factors should be considered in 5
development of an irrigation project? Further, what should be the method
of the valuation and pricing of irrigation water, the optimum size of
the project, the optimum farm size, the optimum farm organization, and
the feasibility of the project as a whole? It would be presumptuous
to assume that all of these questions could be answered here. An
attempt will be made, however, to clarify the economic value of irrigation
water. To achieve this, the conceptualization and;illustration of the na
ture of the water production function will be a necessary intermediate step.
Historically, in the United States, the Bureau of Reclamation has
had the chief responsibility of evaluation of federal investments in
irrigation development. Hence, it plays an important role in the
allocation of vast sums of public capital. Because of the large federal
expenditures,the criteria by which decisions are made becomes rather
critical as far as benefits and costs to society are concerned.
With the demand for public capital as widespread and intense as it
is, the level of accuracy involved in proposed irrigation development
becomes more and more critical. Presumably, the very high "return"
projects already have been built, and, as water becomes more scarce and as society moves to the more "marginal" type projects, the competing demand for funds in other sectors of development requires more refined knowledge and dictates a more critical attitude and more careful scrutiny of the possible value of public investments in irrigation development.
Ascertaining the economic relationships surrounding the water 6
production function will be a contribution to the body of knowledge
and to the advancement of irrigation development through increased
efficiency. This increased efficiency will be due to the greater
certainty of knowledge of the future brought about by the increased
understanding. Farmers and government planners alike will be able to
combine resources and factors more efficiently and an optimum resource
allocation will be more nearly realized.
As one sector of the economy becomes more efficient, the entire economy and society benefit. The benefit is due either to the
reduction of resources necessary to produce a given output of commodities or the greater amount of commodities which can be produced with given resources. The resources not needed can be used in other sectors of the economy to produce other desirable goods.
In the economic analysis of the complex problems of water resources in agriculture, irrigation water can and should be considered the same as any other production factor which takes part in the production process and contributes its share to the total output. In the past, individuals and groups have claimed that, due to its peculiarities, water was unique and required a special set of theory and treatment. When the problem is considered in the most objective manner, it becomes apparent that all factors have uniqueness; the realm of agricultural inputs is really a continuum of elements with "unique" characteristics.
As for any other productive agent in agriculture, the real world problems of the water resource (such as resource development or allocation among 7
alternative uses) extend far beyond the realm of economics.
However, the essence of the allocation problem is economic; consequently,
the attempt here is to show that the conventional models of economic
theory are well suited to the analysis of the major aspects of the
water problem and the provision of economic solutions.
Objectives of study
The scope of the present study is confined to the application of
water as an agent of production in commercial agriculture in the
seventeen western states of the United States. The problem considered
and the frame of reference will be directly related to those particular
areas.
A major concern in considering the economics of irrigation
development is to select those theories which are applicable from the
accepted body of economic theory. In such an investigation, one wants
to ascertain the simplest framework of theory which could serve as a foundation and guide for all planners interested in the allocation of water.
Because of a lack of conformity in the method of water valuation, an attempt is made to ascertain the "best" possible approach to the pricing of factors which-.really do not enter the market place for pricing. In addition, the imputation process of product distribution should be reviewed and improved where possible.
In understanding the problems of the allocation of scarce goods, the determination of the correct relationship and position of the 8
production function in economic investigations will prove invaluable.
The conceptualization and formulation of hypotheses concerning the
water-yield relationship will be a valuable aid in subsequent empirical
research.
Finally, an important objective of the present study is to
illustrate the use of economic theory when applied to the problems of
irrigated agricultural development.
Analytical techniques and procedures
Use will be made of the existing body of economic theory to
formulate models applicable to determining the value of irrigation
water. The theory will be supplemented and extended where feasible.
The accepted principles of soils and agronomy concerning soil-
moisture-plant relations will be drawn upon heavily.
Brief exposition will be made concerning the established budgeting procedures which have been used for many years.
Applications of linear programming will be made to illustrate the
usefulness of this technique in imputing resource values.
Additional models formulated using classical marginal analysis
will aid in the consideration of alternatives for resource valuation.
Plan of study
The plan of study is as follows: In order to properly impute
the economic value of irrigation water, one must know the relevant
production function; to ascertain the correct function requires a 9
knowledge of water-plant relationships; before the water-plant
relationships are meaningful in a production function approach, a
foundation must be established as to the nature of the production
function concept.
Consideration of the role of production functions and outlining
of the theory behind the production function concept will be presented
in Chapter 2. Attention also will be given to the selection and
estimation of functions. This groundwork is necessary as a starting
point in conceptualizing the production function for crops using
irrigation water.
Chapter 3 draws heavily on accepted principles established by
agronomists in an attempt to discover the nature of crop response to
irrigation water. An irrigation scheduling model based upon current water use estimation procedures is developed.
In Chapter 4, existing models of the production response are disregarded, and a quasidynamic model is conceptualized which better represents the relation between irrigation water and yields.
The results of an intensive irrigation experiment are presented in
Chapter 5 for the purpose of economic analyses.
Methods of valuation of irrigation water, both with new and old production models, are considered in Chapter 6.
Finally, a summary is given and suggestions are made for directions of future research and the nature of needs of irrigated agriculture. 10
THE THEORY OF THE PRODUCTION FUNCTION
The firm is a technical unit in which commodities are produced and
the entrepreneur transforms inputs into outputs, subject to the technical
rules specified by its production function. This production function,
a concept in physical and biological science, was largely developed
and mainly used by economists. The entrepreneur's production function
gives mathematical expression to the relationship between the quantities
of inputs employed and the quantity of output produced. A specific
function may be given by a single point, a single continuous or
discontinuous function, or a system of equations.
The production function is defined only for non-negative values
of the input and output levels, with input being any good or service
which contributes to the production of an output, and an output being
a good or service which is desired for consumption by individuals.
The entrepreneur's technology is all the technical information
about the combination of inputs necessary for the production of some
output. It includes all the physical possibilities. Two inputs may
be utilized in various ways to yield a number of different output
levels.
The production function differs from the technology in that it
presupposes technical efficiency and gives the maximum output obtainable from every possible input combination. The best utilization of any
particular input combination is a technical and not an economic problem.
The selection of the best input combination for the production of a lia
particular output level depends upon input and output prices and is
the subject of economic analysis.
Refinements in concepts relating to production functions grew out
of economics probably because of the following reasons:
(1) The nature of the production function is important in economic development and in determining the extent to which national products can be increased from given resource stocks.
(2) The magnitude of production coefficients serve as the base for determining optimum patterns of international or inter regional trade.
(3) The concept is important to certain theories in the functional distribution of income. The conditions under which a total- output can be imputed to the factors from which it is produced with the product just exhausted depends on the nature of the production.
(4) The production function provides half or one of two general categories of the data needed in determining or specifying the use of resources and the pattern of outputs which maximize firm profits.
C5) The algebraic nature of the supply function rests largely upon the nature of the production function.%
Production function defined
The production function or yield response curve is a non-negative mathematical function relating the quantities of inputs employed to the quantity of output produced. It presupposes technical efficiency and provides the maximum output obtainable for the various combinations of inputs, thus constituting a boundary between attainable and unat tainable outputs.
^ E. 0. Heady and John L. Dillon. Agricultural production functions. Ames, Iowa, Iowa State University Press. 1961. p. 1. lib
In its simplest form, yield or output is a function of one variable,
such as water, which is stated by the equation:
Y = f (X^)
where Y is yield and is water applied to the crop. Of course, other
factors are necessary for production. To include additional factors
the function is written:
Y = f (Xj, X2, X3 X^)
where X^ is water, Xg may be plant population per acre, X^ may be
nitrogen fertilizer applied, and all the X's up to X^ are other
independent variables which contribute to the total yield of the crop.
Obviously, thé totality of factors that enter the production function
are not known and may not be finite in number. When one specifies
output as a function of X^ through X^, one explicity assumes factors
X^+2 to Xp are held constant if it is known that p factors affect
production. There may exist other factors for which is not known their
influence on yield. In addition, one might hypothesize that there
exist other important factors which are not yet known. Under continuing
technological progress with the physical relationships of production,
it would be expected that the number of variables in a production function would continually increase.
It is conceivable that there are thousands of inputs which affect crops, some of which are not yet known. These factors which affect production may be divided into four classes: 12
(1) Known factors for which the functions are known but which have very high prices relative to productivity.
(2) Known factors for which the functions are known.
(3) Known factors for which the functions are unknown.
(4) And finally, unknown factors.
All factors that affect production can be fitted into one of the above
classes. Unfortunately, there exist only a few factors in class (2).
Some would fall into class (1) while a great host of inputs qualify
for class (3). Unfortunately there are probably a great many elements
which are found in class (4). However, technological progress in
production is moving factors from class (4) to class C3) and eventually
to class (2). A different type of progress moves factors from class
(1) to class (2). A relevant end of scientific endeavor would be to
maximize the number of factors in class C2).
Type of production functions
There are three general types of relationships which can be
observed in the production of a commodity when inputs are varied. First,
it is possible that the amount of product increases by the same amount
for each additional unit of input. In this case it is said that there are "constant returns" in the output as the input varies in the
production of a particular commodity.
A second type of relationship which can be observed is one in which each additional unit of input results in a larger increase in product than the preceding unit. When this is true it is said that 13
there are "increasing returns" from the input. However, actual cases
of increasing returns in agriculture are not common; and when such cases
are observed, they occur at relatively low levels of inputs.
The third type of relationship which can be observed in production
is one in which each additional unit of input results in a smaller
increase in product than the preceding unit. Thus, it is said that
there are "decreasing" or "diminishing returns." This case is the one
normally expected in the production of agricultural products.
It should be remembered that a production function may include all
of the above three cases in various ranges of input application, and
that the individual factors making up a generalized production
function may exhibit differing qualities when considered separately.
The factors which one might consider for inclusion in a crop production function are of four types. First are the "variable" factors which are those that can be changed at the discretion of the entrepreneur. Examples of variable factors are water, fertilizer, herbicide, insecticide, tillage practices, and seed. These represent the inputs normally under the control of the individual entrepreneur.
fecond are the "correctable" factors affecting yield which are items not presently at optimum levels for crop production but with a cost that can be rectified. These include land leveling, liming, leaching of salts, erosion control, and others.
Third are the "semi-correctable" factors which occur when some inputs can be corrected to improve output but may still exert a 14
depressing effect upon yields. Examples are the removal of prohibitive
stones, land drainage and improvement, and, to a certain extent, soil
texture made more favorable.
Fourth are the "non-correctable" factors which cannot be changed
but do exert an influence upon production. The most obvious of these
are the weather, geographical location, depth of soil, and parent
materials of soil. It is desirable to know how each factor affects
yields and what, if anything, can be done about deficiencies.
It is apparent that the above factor types are not hard and fast
rules and that many borderline cases will occur where either of two
possibilities seem equally appropriate. The classification exists only for the convenience of discussion. The characteristics of factors are ascertained by the physical scientist and as such provide the economist with the basic information needed for the estimation of production functions. Hopefully, as time goes by, progress will continually be made in the specification of factor characteristics, and the advance of technical knowledge will make possible a shift of individual factors to a more useful class, such as from semi-correctable correctable.
Physical production functions
The nature of the fertilizer-crop production function has been a subject of debate for generations, and it seems reasonable that no single form can be used to characterize agricultural production under all environmental conditons- Therefore, a number of alternative 15
algebraic forms may be used in estimating production functions. The
algebraic form of the function and the magnitude of its coefficients
will vary with soil, climate, type and variety of crop, inputs being
varied, state of thearts, and other variables. The selection of any
specific type of equation to express production phenomena automatically
imposes certain restraints and assumptions with respect to the
relationships involved and the optimum input quantities which will be
specified. It is therefore necessary to select a functional form which
appears or is known to be consistent with the phenomena under investi
gation. An infinite number of functional forms are possible, but only a few of those with logical implications are considered here.
To acquire any output at all, more than one factor of production
must be used. One needs at least two inputs for an output to be forth coming, and most single input functions assume some underlying factor as fixed.
Typically, certain categories of inputs are held at a fixed level while others vary. Some factors are exogenous to the system and are not subject to the control by the entrepreneur. The exogenous variables are regarded as random disturbances or error. The production function may then be expressed as:
Y = f (X^, , ..., X^) + e where e is an error term. If only one factor is allowed to vary, a bar is inserted:
Y = f (X^l Xg, ..., X^) + e 16
All factors to the right of the bar are held or assumed constant.
For simplicity one usually writes:
Y = f (Xj)
which implicitly assumes all else is held constant.
Cobb-Douglas function One of the most popular functions used
of recent years is the Cobb-Doublas or power function. The most used
form is:
Y = aXb where X is the level of input allowed to vary, Y is the physical out put, a is a constant and b is the transformation ratio. The power function allows increasing, constant, or decreasing returns; but no more than one at a time is allowed. The exponent, b, defines the elasticity of production and is constant over all ranges of output.
The Cobb-Douglas function cannot satisfactorily be used where one finds ranges of both increasing and decreasing returns. As input increases, a maximum total physical product is never reached. The equation assumes the input to be limitational such that when input is zero, no output is forthcoming.
When more than one input is allowed to vary, the function takes the form:
Y = aXi^lX2^2 ...X^^n when n variables are considered. In this case the sum of the exponents gives the elasticity of production. Once again each input is limitational; no output is produced if any input is at a zero level. No maximum out put is defined as in the case of one factor. 17
Spll"'inan function The function suggested by Spillman may be
considered in the form:
Y = M - A
where x is the level of input, Y is the resulting output, M is the
maximum output which can be attained, A is the total increase in out
put due to X, and R defines the ratio of successive units of input
to total output. A maximum product is never reached and decreasing
total product is not allowed.
Quadratic forms A quadratic in one variable might be
represented as: 2 Y = a + bX + cX
where X is the level of input; Y is the resulting output; and a, b, and
c are constants of the equation. A maximum total physical product
is defined and the function allows both decreasing returns and
diminishing total output ranges. An extension of the quadratic form to include two variables may be represented as:
Y = bg + b^X^ + ^^2^2 ~ ^11^1 ~ ^22^2 ^ ^12^1^2 with the definitions as above. Either of the above may be modified to give a compromise between the power function and the quadratic form.
The compromise is a square root equation such as:
Y — SL — b^X^ — ^2^2 ^3^1* ^ bgX^ X2
The quadratic and square root forms provide an interaction term between the two varying inputs. As the number of inputs increases, the number 18
of interaction terms increases accordingly. If the investigation implied
a function which has ranges of both increasing and decreasing returns,
the function
Y = a + bx + cx^ - dx^
will allow this condition.
Derivations of production functions
From the production function derivation of various quantities
may be made which are useful for economic and other analyses.
Once the functional form is specified the mathematical derivation
of the equations for the marginal rate of substitution,isoquant s, isoclines and profit maximizing level of input use is straightforward. Examples will be shown for the quadratic function.
Marginal rate of substitution The marginal rate of substitution of one input for another is defined as the amount of one input that could be replaced by one unit of the other input while the output remained constant. The marginal rate of substitution can be evaluated as the ratio of the marginal physical products; the marginal physical product being the partial derivative of the production function. For the quadratic function MPP ^ bg - 2b22 Xg + b^2 *2 ^ specifies the marginal rate of substitution.
Isoquants The marginal rate of substitution specifies the slope of the isoquant. The slopes of each isoquant indicate the rate 19
at which one input will substitute for another. Any output level on
a production surface is gained by a particular combination of inputs. For
a given level of output, any number of combinations of inputs ca» bz used.
The equation which depicts the alternative combinations of inputs to
achieve a constant yield is called an isoquant. The isoquant equation
is:
X =^1 ^^12^2 -[(^1 + ^ + ^22^2 " ^2^2 ~ ^o^^^ 1 2 bii
which is derived by solving the quadratic function for one of the
inputs.
Isodines An isocline connects points of equal marginal rates
of substitution on successive isoquants and thus is useful for economic
analyses.
The consideration of isoquants indicated that it is possible to achieve a given yield with many combinations of inputs. If the price of the inputs is known, the least cost combination to obtain a given yield can be determined. The least cost combination is derived by equating the price ratios to the inverse of the marginal rate of substitution.
J PXg <5^ = px^
Moving along any isocline indicates the least cost combination of inputs to use to obtain varying output. The isocline equation is derived by setting the marginal rate of substitution equal to a constant input price ratio, k. We have then 20
kbi - bg kb^2 + 2^22 = 2kb 11 * ^12 '*»11 " ^2J
as the isocline equation for the quadratic function.
Profit maximization For any given factor price ratio the profit
maximizing level of input use can be specified. Profit maximization is
achieved by equating the value of the marginal product to the input
price. For the quadratic function we have
"y \ and
The two equations must be solved simultaneously unless there is no interaction term in the function. The simultaneous solution yields
-1 (2b22 Pjf + ^12 -X ) ^y - ^22 + h ^12) ^1 = _ 4b 12 11 22 and
-1 «"11 'x. h2 - (2bjj bl;) '=2 = bi2 - 4bii ^22 where the level of input use for profit maximization is a function of the constants of the equation and the prices of inputs and output.
These equations provide an input demand function in which one can trace the changes in input use as prices vary. 21
Appropriate functions
The appropriate form of function to be fitted to a production
surface should be selected in terms of the environment and nature of
the unit for which it is estimated. A fertilizer production function which includes both applied and soil nutrients in the variables would
have characteristics such as those represented by the Square Root
function or perhaps the Spillman function if a range of decreasing total product is not expected. If the soil test of nutrients was
included in the function, the most applicable function might be the
Quadratic form.
The appropriate functional form for estimating input-output relationships will depend upon the type of phenomena under investigation.
Usually a crop response to fertilizer will have diminishing returns for all positive input levels. Under most circumstances, a function which allows decreasing total physical product would be needed.
Eblection of the form of the function
Different research workers emphasize various methods in selecting the form of the function-to be used. At one extreme, one may prefer to start with a hypothesis that the underlying production function is of a certain algebraic form. This form would be invariant and the parameters would be estimated by observed data.
Other workers would prefer the selection of a polynomial and estimate the coefficients, retaining only those variables with statistically significant coefficients. 22
A third approach is to make no assumptions or hypotheses
and simply to use the form of the function which best "fits" the data
by statistical analysis.
One is not restricted to the use of only one of the above
approaches but may find it profitable to use some combination of the
three.
Another alternative to use in deciding between various functions
is to compare predicted and observed yields over the relevant range
of the production surface. This is very appealing when used in
conjunction with alternative methods. A refinement of the estimated coefficients and the functional form itself may be possible by comparisons after the "best" form and parameters have been estimated. 23
WATER-PLANT RELATION ailPS
Water is an essential nutrient for plant growth, and it must
be supplied in larger quantities than any other nutrient to satisfy the évapotranspiration requirements of growing plants. Unlike other nutrients that are absorbed and retained by the plant, water is not retained but has a continuous one-way flow from the soil through the roots, up the stems, and into the leaf surfaces, where it is evaporated mainly from inside the stomata and diffused into the air.
The main uses of water by the plant , in addition to actual use in the individual cells, are controlling the temperature and translocating the nutrients from the soil to various parts of the plant itself. Water also acts as a solvent which together with the dissolved nutrients make up the soil solution. Moisture is held in the soil with suction or tension and work must be done to remove this water. The tension with which the water is held depends upon the amount present.
With irrigation, the farmer can exercise a much greater control over soil moisture than over any other physical soil factor- At the time of irrigation, the soil is essentially filled with water because all the pores excepting those containing trapped air are filled with water. After twenty-four to forty-eight hours the profile has drained
* to field capacity. At this stage the water has drained from the larger
* Field capacity is defined as the amount of water retained in a soil against the force of gravity at any specified time after flooding. It is approximated by one-third atmosphere. 24
pore spaces but remains in the smaller pores. The outer surface of the
film of water on the soil particles is held with a tension of about
one-third atmosphere.
As water is extracted by the plants the thickness of the water film
decreases and, as the film decreases, the tension by which the water is
held to the soil particles increases. The removal of water from the
soil will continue as long as the osmotic forces within the plant
cells exceed the retentive forces of water held on the soil particle
surface. When the rate of removal is too slow to maintain plant
turgidity, permanent wilting occurs. At permanent wilting the average
moisture tension at the outer surface of the moisture film is about
fifteen atmospheres.
Figure 1 shows the various stages of moisture tension and the
thickness of the moisture film on the soil particle is illustrated in
Figure 2.
The general relationship between soil moisture characteristics and soil texture is shown in Figure 3. The field capacity increases until silt loam soil is reached. Figure 4 indicates the variations in the
movement of water through the soil at irrigation due to soil differences.
In addition to the soil texture characteristics, other factors may influence the water holding capacity of soils, such as the amount • 25
ytiVi ré 'i!//f;'i'j/M\
Saturât ion Field capacity Wilting point
Pore space ^
".wat&r ' Saturated soil
a I r Field capacity
• a ir Wilting coefficient
BM' • .
ai r Hygroscopic coefficient IB'
rigure 1. The volumes of solids, water, and air in a silt loam soil at different mois lure levels (Redrawn from H. 0. Buckman and N. C. Brady. The nature and properties of soils. 4th ed. New York, N. Y., The Macmillan Co. 1963. p. 164.) Soil-water Hygroscopic Wilting Field interface coefficient coefficient y
I 10,000 ! 31 15 atmos. ' atmos. atmos, atmos. I Flows
gravity/
10,000 atmos, tens ion
c o (/) c 0)
1/3 atmos. tens ion
very thin Increasing very thick film moisture f iIm
Figure 2. The relationship between thickness of water films and the tension with which the water is held at the liquid air interface (Redrawn from H. 0, Buckman and N. C. Brady. The nature and properties of soils. 4th ed. New York, N. Y., t he Macmillan Co. 1963. p. 168.) :'7
30
24 -
w 18 - (D 3 c 0) u •- 12
V Unavaiiabie water
6 -
sand sandy I oam silt 1 oam I oam increase in heaviness of texture
Figure 3- The general relationship between soil moisture characteristics and soil texture (The wilting coefficient increases as the texture I'ecomes heavier. The field capacity increases until silt loam is reached and then levels off.) (Redrawn from H. 0. Buckman and N. C. Brady, The nature and properties of soils. 4th ed. New York, N. Y., The Macmillan Co. 1963. p. 176.) Sandy loam Clay loam
0
12
2k
1* hou 48 hours 48
,24 hour 60
72
Distance—inches from center of furrow
Figure 4. Comparative rates of irrigation water movement into a sandy loam and a clay loam soil (Redrawn from J. J. Goony and J. t. Pehrson. Avocado irrigation. University of California Extension Leaflet 50. 1955.) 29
of organic matter, the strength of the osmotic effects, the total pore
size and pore size distribution, and the depth and type of soil
profile.
In the Netherlands, Visser^ has done considerable work relating
crop growth to the availability of moisture. His results strongly
support the view that the soil moisture tension is a basic factor
for plant growth. The plant has to compete with the capillary
effect of the soil, and the nearer the two capillary forces of suction
by the plant and of retention by the soil approach each other, the
lower the yield will be.
Factors affecting moisture availability
It may be well to consider some of the more important findings
made by agronomists and soil physicists regarding soil-water-plant relationships, particularly considering the pattern of water extraction by plants and the availability of water to plants from various soil types. Most irrigationists recognize today that the relationship is not a fixed one, but may vary under different conditions. (&e Figure 5.)
A significant difference of opinion remains regarding the pattern of water extraction by crops between field capacity and the wilting
^W. C. Visser. Crop growth and the availability of moisture. Institute for Land and Water Management Research, Wageningen, Netherlands, Technical Bulletin 6. 1959. 100% Available moisture depletion, percent
Figure 5. Variations in how relative growth relates to available moisture depletion for sandy, loam, and clay soils (Redrawn from R. H. ilagan. Water-soil-plant relations. California Agriculture 10, No. 4: 9. April, 1957.) 31
point. Because various soils retain different amounts of water at
field capacity and wilting point, they therefore have different water
capacities. Some irrigationists contend that the water held by the
soil is equally available for growth until very near the wilting
point. However, others maintain that plant growth diminishes pro
gressively as the soil moisture content falls below field capacity and
that growth ceases at the wilting point.
Because the type of experiments to be conducted for establishing
production functions for water are determined by the school of thought
which is adopted, it becomes important to ascertain in advance of any
production function experimentation the availability of water to
plants.
There is really very little direct information on the relation of
plant growth to soil moisture stressj for, except by irrigating to
limit the maximum stress value attained, there is no known method of
controlling soil moisture stress during plant growth. Recent information
suggests that the vegetative growth of some plants decreases significantly
as the soil moisture stress increases in the range of 1 atmosphere and
becomes zero before reaching a soil-moisture stress of 15 atmospheres.
This is out of harmony with the previously held view that soil moisture
in the so-called available range is equally available for plant growth.
It is essential that the evidence be closely examined in relation
to the effect of variation in soil moisture within the available range growth response. To evaluate the evidence provided by a given 32
experiment involving variation in soil moisture supply, it is important
that consideration be given to (1) the nature of the soil involved:
C2) the stage of plant growth being studied, including the criterion
of growth response; and (3) the climatic influences.
Soil factors The texture, structure, and depth of the soil in
the root zone determine the capacity of the soil for storing the water available for plant growth and are very important in determining the
change in soil moisture stress taking place with time and the change
in moisture content as the plant withdraws the water from the soil.
Any soil factor which affects root density can be expected to
influence the response of a plant to irrigation. Mechanical impedance, slow"water penetration, poor internal drainage, and deficient aeration frequently are responsible for sparse and shallow roots.
Salinity may affect the soil moisture - plant growth relation by decreasing moisture availability through increased soil moisture stress, by interfering with root growth and absorption through toxicity reactions, and by contributing to poor soil structure. The poor soil structure in turn influences infiltration, drainage, aeration and root growth.
One may expect the interaction between fertility and water to also have a determining effect on the growth response of crops to irrigation. Fertility responses have complicated the interpretation of many experiments on soil moisture versus plant growth.
Plant growth stage and growth response criterion Three distinct stages of growth should be considered when contemplating moisture 33
response effects. The first stage of growth is the period of germination
and establishment of self sufficiency of the plant. The second stags
is the period of vegetation, the most important stage as far as time
involved. The third stage is the period of fruiting and maturation.
However, in the culture of some crops, the third stage is missing because
the plant is grown for its vegetative output and not the reproductive
parts; in rare instances, one is concerned only with stage one.
It is possible and highly probably that the plant responds
differently to water in these three regions or stages. For example
a very wet treatment that optimizes yields during vegetative growth may
be reversed during maturation in order to provide a greater yield.
The effects of given soil moisture stress conditions on crops,
then, are often dependent upon the stage of plant growth.
When considering plant response it is necessary to agree on an
appropriate measure of plant growth. There are at least four measures
of growth that have been suggested. First, the elongation of plant
organs, whether it be of leaves, branches, or stems, has been used
rather commonly, Sacond, an often appropriate measure is weight of the
plant using either fresh or dry weight depending upon the type of
plant. Third, the amount of carbon dioxide used by the plant is a valuable measure to use in the green house, although it would not be appropriate for field experiments. Fourth, another measure of plant activity and growth which has been used on occasion is photosynthesis-
The above measures of plant growth, unfortunately, are not all 34
equally affected by increasing soil moisture stress. Therefore, the
use of various measures may indicate various moisture response.
It appears that photosynthesis and respiration are insensitive
to moisture stress. Dry weight is more sensitive, but fresh weight and
elongation of plant organs seem to be the most sensitive to changes
in moisture availability.
Not all crops respond to moisture availability or tension in the
same manner, so the difference between crops themselves would be
significant.
The nature of the root system is, of extreme importance in
determining the relation between measurable soil moisture stress and
plant growth. The fraction of the available moisture range which can
be utilized before growth is checked will vary with root density.
Climatic factors Weather conditions may influence the
growth characteristics of shoot and root development and affect moisture-
plant growth relations. Air temperature, sunshine, humidity and wind
movement influence the rate of water loss by transpiration from plant
leaves and evaporation from the soil surface. All climatic influences
have an important modifying effect upon the efficiency with which the
water in the soil reservoir will be used in plant growth.
Plant response to water
To what extent can the soil moisture supply be reduced without causing permanent injury to the plant? 35
The data in Figures 6, 7 and S emphasize that not all water
between field capacity and wilting is equally available to the plant.
It is to be noted for Figure 8 that, although growth ceased as
the wilting range was approached, it increased rapidly again almost
immediately following irrigation- Whether or not this affected the
yield of cotton is another matter.
For crops where the yield is measured in terms other than vegetative growth, one may obtain sizable increase in leaf and stalk growth main taining the moisture level well above the wilting percentage without an accompanying increase in yield. More information is needed to establish not only the relationships between vegetative growth and moisture stress but also the relationships between yields of seed and moisture levels in the soil.
More than any other factor the type of soil causes the amount of water required by a given crop to vary. Although topography, the nature of the soil profile, and the texture of soils are taken into account in a broad way to establish the desirability of land for irrigation, assuming that fertility and salinity can be remedied, there is a need for laboratory evaluation of soils to obtain a more adequate picture of their moisture status. The moisture tension curve is an excellent means of providing such information.
It is significant to observe that the various soils have different amounts of available water at the same tension. en
Réchappa fine sandy loam
Hanford sand
2 3 15
Tension, atmospheres
Figure 6. Moisture retention on soil core with natural structure (Redrawn from L. A. Richards and C. H. Wadleigh, Soil water and plant growth. Agronomy 2: 73. 1952.) 37
Moisture stress in atmospheres
Figure 7- Growth of bean plants as influenced by the soil moisture stress (Redrawn from C. H. Wadleigh and A. D. Ayers. Growth and biochemical composition of bean plants as conditioned by soil moisture tension and salt concentration. Plant Physiology 20: 124. 194-5.) 38
leaf 5 120 leaf 6
eaf 7
100
cn
20
irrigations
8/7 8/12 8/17 8/22 8/27 9/1 9/6
Date
Figure-8. Elongation of cotton leaves in relation to irrigation (Redrawn from C. H. Wadleigh and H. G. Gauch, Rate of leaf elongation as affected by the intensity of the total soil moisture stress. Plant Physiology 23: 488. 1948.) 39
Moisture movement through the soil at irrigation
After irrigation the direction of water movement is independent
of texture differences in soils, but the rates of movement vary
greatly, as shown in Figure 4. The flow pattern is symmetrical at
all times out from the source of the flow.
The flow pattern of water through the soil is shown in Figure 9.
This assumes a homogeneous profile with no abrupt changes in texture.
The method of water movement through the soil is dependent upon
the type of soil considered. In a sandy soil the wetted perimeter is
advanced through the action of gravity. In a fine soil such as clay,
the capillary action is more important than the gravity action, so the
wetted perimeter is advanced by capillary action. Somewhere between
the extremes of pure sand and pure clay exists a combination of gravity
and capillary forces which are equal in effect. One might think of
this as a continuum as shown in Figure 10.
The movement of water in a dry clay soil would be dominated by
capillary forces, and the pattern of moisture distribution would be
expected as shown in Figure 9. As the time irrigation progressed the
boundaries would move but at any one time there should exist a four
region classification of (1) free water (above field capacity), (2) field
capacity, (3) adhesive and capillary water, and (4) dry soil at
permanent wilting point. Of course, if the soil profile had been wetted previously, the dominating forces might have been entirely different, depending upon the residual moisture content at time of
irrigation. Dry soj1 • wat
Wetted perimeter
Figure 9. Pattern of flov/ of water through a hypothetical homogeneous soil profile Gravity water Gravity and capillary Capillary water dominating equal in effect dominating
Sands Clays Type of soil structure continuum
Figure 10. Effect of soil upon tine type of water movement in a profile 42
Plant growth response
The growth of most crops produced under irrigated farming is
stimulated by moderate quantities of soil moisture and retarded by-
excessive or deficient amounts. A certain quantity of air in the soil
is essential to satisfactory crop growth; hence, since excessive
flooding and the filling of the soil pore spaces with water drive out
the air, the proper functioning of the plants is inhibited eventhough
an abundance of available water is supplied. Soils having deficient
amounts of water hold it so tenaciously that plants must expend extra
energy to obtain it; if the water supply slightly decreases even
further, the moisture content decreases until the rate of absorption is
not high enough to maintain turgidity, and permanent wilting follows.
At some moisture content between these two extremes it has been thought
that plants grow most rapidly. (Sfee Figure 111)
Because of the wide variability in the physical properties of
different soils, the moisture percent at which a plant wilts permanently varies considerably. Extended investigation by Briggs and Shantz^ led to the conclusion that nearly all plants wilt at substantially the same moisture percentage in a particular given soil.
Of greater interest to economists is the nature of the moisture relation curve. One theory, the "equal facility" school of thought
^L. J. Briggs and H. L, Siantz. The water requirements of plants. United States Department of Agriculture, Bureau of Plant Industry Bulletin 284. 1913. Hygroscopic Wilting Moisture zone Water moisture zone of optimum saturat ion zone growth zone
very dry > \ very wet
Moisture content of soil
figure 11. Rate of growth in relation to the moisture content of the soil (hypothetical) 44
argues that plant growth is substantially the same in the range of
optimum moisture. Another theory, the "inverse function" school
of thought, argues that the growth rate of crops declines continuously
as the moisture content changes from field capacity to the permanent
wilting point.
The "equal facility" theory A study conducted in California
by Hendrickson and Veihmeyer^ concluded that plants extract the soil
moisture necessary for ths ir continued growth equally well between
wilting percentage and field capacity. Water is equally available
between these two points because the force with which soil particles
hold it changes little between field capacity and the wilting point.
Once the neighborhood of wilting point, is reached, the soil moisture
tension increases rapidly and further growth is seriously retarded.
The hypothesized relation between soil moisture and plant growth could
be characterized as in Figure 12.
Wilting frequently occurs at much higher moisture levels (lower
stresses) than the permanent wilting percentage. Any part of a plant
may wilt when the transpiration rate exceeds the rate at which the water can be replaced. If other factors remain about constant, temporary or transient wilting will increase in plants as the soil moisture approaches the permanent wilting percentages.
^A. H. Hendrickson and F. J, Veihmeyer, Irrigation experiments with grapes. California (Berkeley) Agricultural Experiment Station Bulletin 728. 1950. 45
Soil moisture
Figure 12. Hypothesized soil moisture-plant growth relationship under the "equal facility" theory 46
The physical tension of soil moisture is important in determining
the ease of water absorption by plants. The complete picture of
water availability in soils, however, must recognize that soil water
is not pure. It dissolves various materials from the soil, and so,
like the plant cell sap, it possesses a definite osmotic pressure
due to dissolved ions. In many cases the osmotic pressure of these
soil solutions may be equal or greater in stress than the physical
tensions as shown in Figure 13. In view of such effects of salt, it
is apparent that any attempt to control soil moisture without
recognizing osmotic pressure of the soil solution might fail entirely.
For many soils the available moisture capacity increases as the
textural grade becomes finer. There may be exceptions to this however,
because some heavy soils contain appreciable amounts of water at the
permanent wilting percentage.
The significance of field capacity as the upper limit of available
moisture may be less in irrigated soils than under many other conditions.
The soil water is equally or more available to plants above field
capacity than at or below field capacity. Since noncropped soils have
been found to take from 48 hours to six days to reach field capacity following saturation, plants may have time to utilize appreciable
quantities of water above this level. This use of water above field capacity would be further increased by frequent irrigations, even though the period between the end of irrigation and the reduction of moisture to field capacity would be greatly shortened by the growing crops. 47
* 1 I- 1 — —
0) L. 0) -C o. 15 tn O m • c l\\ v> VÎ 1 .2% salt O tn m o 1% salt salt 1 1 t 1 10 15 20 Percent soil moisture Figure 13. Relation of moisture stress to moisture percentage at different levels of soil salinity (Redrawn from C. H. Wadleigh and A. D. Ayers. Growth and biochemical composition of bean plants as conditioned by soil moisture tension and salt concentration. Plant Physiology 20: 122. 1945.) 48 The optimum level of moisture for plant growth is especially im portant in irrigation agriculture because this optimum is fundamental in determining the best time for water application. The "inverse function" theory Wadleigh, Gauch, and 0. C. Magistad conducted a study of growth of guayule in relation to the total stress of soil moisture. They found that the growth of the plant is inversely re lated to the soil moisture tension. As that tension increases, growth de creases.^ Wadleigh attributes part of the conflicting conclusions about op timum moisture for plant growth to the hyperbolic nature of the relation ship between soil moisture percentage and total moisture stress. As the quantity of water is decreased by the grcwing plant, stresses on soil moisture resulting from increased concentration of salt in the soil solu tion are likely to increase much more rapidly over the available range than the stresses due to physical forces. It is also apparent that plants may suffer from water unavailability in saline soils even when the soils appear to be quite moist. Some of the differences between the two conclusions concerning the optimum moisture for plant growth is probably due to climatic conditions. At tensions above one atmosphere, the amount of available water in a soil is often only a small fraction of that present at field capacity. The moisture is present in thin films, and though ^C. H. Wadleigh, H. G. Gauch, and 0. C. Magistad. Growth and rubber accumulation in guayule as conditioned by soil salinity and irrigation regime. United States Department of Agriculture Technical Bulletin 925. 1946. 49 available to plants can be absorbed at only slow rates. During periods of rapid transpiration, plant tissues are partially depleted of water, wilting results, and growth is retarded. Further field studies are needed before the optimum moisture availability for each stage of plant growth can be determined. Tests need to be conducted on a variety of crops and soils. Solution to problems such as aeration and availability of plant nutrients may show non-direct factors which will inhibit growth. Although the problem of optimum soil moisture for plant growth has not been completely solved, many of the points of apparent disagreement are of little significance in irrigated agriculture. Because the relative humidity is usually low and the transpiration rates are high, the last portion of available moisture cannot be safely relied on if the soil moisture remaining in the soil above a tension of one or two atmospheres is comparatively small. Therefore, with most crops it is time to irrigate when the soil moisture tension in the root zone approaches one atmosphere. The total water requirement varies between crops, as does the time of maximum demand. Most agree that the yield of crops in relation to water follows about the same general relationship as has been established between yield and plant nutrient. (&e Figure 14.) In relation to water and plants, it is apparent that if there is no available water there will be no growth. As units of water are added, growth will increase rapidly but at a diminishing rate due to other factors being fixed. One would expect a maximum yield point beyond 50 ! I Maxima 1 yie1d — — — —] / low/ Available water in the soi 1 high Figure 14. Curve showing general relationship between the content of available water in the soil and the yield of crops 51 which further addition of water will decrease total yields- It is evident that attention to only total water applied may ignore the fact that during certain critical periods plants may suffer from drought and that in some cases a "stress" on the plant at a particular stage of development may increase yields. Therefore, the timing of the application of water may be much more important than the quantity. Also, one must not ignore the principle that fertilizer and water requirements of a crop on any soil must be defined in terms of each other. The idea of relating plant growth first to moisture tension and only later and indirectly to water quantity has been a great step forward for the agronomist who is interested primarily in finding the general relationship between moisture conditions and plant growth. The introduction of the concept of moisture tension has enabled the agronomist to construct a relationship which is in a sense independent of soil type and water quantity. Once this general relationship is known, it is relatively simple to determine, on the basis of known moisture stress curves, the water quantity required to fill a particular soil to field capacity. From the economic viewpoint, unless water is a free good, optimum application would be to keep the moisture level at some point between field capacity and the wilting point. However, it is impossible to wet any soil mass to less than field capacity. If a small quantity of water is applied to the mass of dry soil, the uppermost layer is filled 52 to field capacity while the balance of the soil remains unaffected. As more moisture is added, the soil is wetted to a greater depth. The wet layer will be separated from the dry layer below by a rather sharp line of demarcation. The moisture content of any particular soil layer can only be reduced through extraction by plants or evaporation. This framework is a static one only and does not consider time as a variable. Theoretically, a given quantity of water can be distributed through the season in an infinite number of ways. With a certain quantity of water, some time distribution will produce a greater product than others. The fact that most soils and crops require irrigation water to be applied at intervals throughout'the season explains the importance which irrigation scientists have attached to the•determination of that moisture percentage which constitutes the wilting point. Integrated moisture stress Unfortunately, the moisture content of the soil does not change uniformly after an irrigation. There are two reasons for this lack of uniformity- (1) The moisture of the upper layers of the soil profile will be reduced more quickly to the wilting percentage through evaporation, while considerable more moisture will remain in the lower layers. (2) In those parts of the soil profile which contain the 53 heaviest concentration of plant roots, soil moisture tension will increase more rapidly due to extraction by the plant. This problem is aggravated by the absence of effective moisture movement in the soil. Thorne and Peterson point out that "the growth of roots is generally considered more important than water movement in bringing new supplies of water to the plant. Admittedly, some moisture movement occurs, but it is quite negligible. As Richards and Wadleigh stated, "The effective distance through which water in the available range can move toward the root is certainly of the order of inches and not feet. The pattern of moisture extraction is therefore largely a matter of the active root distribution. Root distribution is mainly determined by the genetic character of the plant but is modified by plant spacing as well as by soil and climatic 2 factors It follows then that the soil moisture condition which generates wilting is not of single point but rather a certain distribution of soil moisture throughout the profile. Therefore, Beringer concludes that: (1) within any one layer of soil, moisture tension will be an 1 D. W. Thorne and H. B. Peterson. Irrigated soils, their fertility and management. Philadelphia, Pa., The Blakiston Company. 1949. P. 39. ^L. A. Richards and C. H. Wadleigh. Soil water and plant growth. Agronomy 2:92. 1952. 54 increasing function of time; and (2) at any one period of time, moisture tension within the soil profile, between layers, will be a decreasing 1 function of depth. 2 3 Wadleigh and Taylor have suggested methods of combining these distributions into a single number by means of a concept which they have called "integrated moisture stress." Wadleigh's method The total soil moisture stress S at any one point in time is expressed as a linear function of the soil moisture tension T, and the osmotic pressure of salts in the soil as^ S = T + T is a decreasing function of depth and ^ is an increasing function of depth due to higher accumulations of salts in the lower stratas. It is hypothesized that these two forces tend to effect one another so that the moisture stress is somewhat uniform throughout. The assumption is made then that S is not a function of depth any longer but a function of time. S is then integrated over time to obtain the integrated atmosphere days, A. ftX A = Ito Sdt ^C. Beringer. An economic model for determining the production function for water in agriculture. California Agricultural Experiment Station Research Report 240. 1961. 2"C. H. Wadleigh. The integrated moisture stress upon a root system is a large container of saline soil. Soil Science 61:225-238. 1946. 3 S. A. Taylor. Estimating the integrated soil moisture tension in the root zone of growing crops. Soil Science 73 :331-3 40. 1952. 55 The integrated atmosphere days are the sum total of the different stresses exerted each day during the irrigation cycle, t^ represents the number of days in the cycle and A/t^ represents the average soil moisture stress during the interval. Either A or A/t^ can then be used as the independent variable in determination.of a production function. Taylor's method This method is more general given as: T = F Cx, t). Tension with respect to depth is expressed as a function of x: T. = f(x) 1 with respect to time: Tj = gCt). By combining the two equations: F.. = F(x, t) the double integral formed is: szi d X dt where Tr equals integrated moisture stress in root zone. If the soil mass which is to be wetted is held constant, then variation in integrated moisture stress which results from more frequent irrigation will be related much closer to labor and capital inputs than to water quantity per se. Significant variations in water quantity will be encountered only when we introduce variations in the depth to which a soil mass is to be wetted. The most interesting approach to the problem of variations in moisture content throughout the soil profile is that outlined above by Wadleigh where use is made of the effect of salts in the soil solution 56 upon the availability of water. With repeated irrigations the salts are continually leached downward in the profile. One could then hypothesize that the concentration of salts varies directly as the depth of the profile varies. Assuming the loss of moisture from the soil to vary inversely with depth, one could conclude that the water in the soil is equally available to plants throughout the profile. Generalized relationship Using the integrated moisture stress concept to account for the heterogeneity of the distribution of water through the soil profile, a generalized yield relationship can be derived. Letting S represent the moisture stress condition and Y be yield we can form: Y = f (S-1) which shows yield a function of the inverse of moisture tension. The derivative of the above function would show the rate of increase in the total output resulting from a reduction in moisture stress conditions between irrigations. The value for S may be determined by taking (1) one moisture stress reading at some point in time, (2) as the mean [—^)of a number of readings, or (3) as the sum of daily readings (Z S^). It matters little i which is used as long as one method is used consistently. For purposes of economic analysis, S ^ is only an index and does not in itself have a price which could be compared with the marginal value product. In order to determine an economic optimum it would be necessary to express S"^ as a function of total water quantity and other 57 associated inputs. It will be observed that as the maximum moisture stress conditions which are allowed to occur between irrigation decrease, the irrigation frequency rises. Associated with higher irrigation frequencies are somewhat larger total water inputs and considerably larger inputs of labor and capital. Only when the combined cost of these factors, which are associated with each value of S~^, have been ascertained will it be possible to establish for each of various soil types and crops those irrigation frequencies which are economically optimum. Irrigation scheduling model Considerable research has been conducted over many years to calculate the "water requirements" of crops. Most investigations follow the "equal facility" school of thought. Significant work has been done by Dowry and Johnson^; Penman^; Thornthwaite^; and Jensen and Haize^. Probably the most widely used is the formula developed by Blaney and Criddle^. ^R. L. Lowry.and A. F. Johnson. Consumptive use of water for agricul ture. American Society of Civil Engineers Conference Proceedings 67:595-616. 1941. 2h. L. Penman. Estimating evaporation. Transactions of the American Geophysical Union 37: 43-46. 1956. 3 C. W. Thornthwaite. An approach toward a rational classification of climate. Geographical Review 38: 55-94. 1948. ^M- E. Jensen and H. R. Haize. Estimating évapotranspiration from solar radiation. Journal of Irrigation and Drainage 89: 15-41. 1963. ^H. F. Blaney and W. D. Criddle. Determining consumptive use and irri gation water requirements. United States Department of Agriculture Technical Bulletin 1275. 1962. 58 Blaney and Griddle developed an empirical formula using climatological data and past research consumptive use data. Originally it was used for computing seasonal use but has now been applied for short-period use. The Blaney-Criddle formula U = KF was first developed to determine the seasonal coefficient "K" where U = consumptive use and F = the sum of monthly consumptive use factors. Consumptive use is considered to vary with temperature, daylight hours and available soil moisture. Multiplying the mean monthly temperature, t, by the possible monthly percentage of daylight hours of the year, p, gives a monthly consumptive use factor, f. The sum of the monthly "f" factors gives the seasonal "F" factor. If consumptive use is measured at a location for a specific crop and the "F" factor can be determined for that year at that location, then the "K" factor can be calculated. It is this "K" factor that is transposed to other areas using the local area "F" factor, from which estimates of consumptive use are made. Consumptive use is affected by many factors. The most important natural factors are climate, soil and topography. Climatic factors include precipitation, solar radiation, temperature, humidity, wind and length of growing season. All of these factors may influence plant growth and thus consumptive use. The most common methods of determining the amount of water consumed by crops are: (1) Measuring the amount of water necessary to maintain 59 satisfactory plant growth in ly^imeters; C2) Measuring the quantity of water applied to field plots; and (3) Determining the soil moisture depletion by soil sampling. Irrigation water requirements are a function of consumption use, which represents the amount of water needed by the plants being irrigated- Part of this water can be supplied by water stored in the soil, rainfall, and a high water table. The stored water, rainfall, and ground water contributions are subtracted from the consumptive use requirement to determine the amount of irrigation water to apply. To test the usefulness of the consumptive use approach, a computer model was formulated. Using climatological data, short-period water use was calculated and an accounting kept of the soil moisture reserve. The period used was one day. Thus each day consumptive use was calculated and subtracted from the available soil moisture. In this manner the amount of moisture available for plant use was always known. As an example cotton grown at Mesa, Arizona was used. Each day the calculation was made of the expected use for the next five days based on historical climatic information. Then, the farmer would have sufficient notice to get water on his crop before the critical time arrived. Four criteria were set up to determine irrigation need for comparative purposes. The criteria were (1) irrigate when 100 percent of available soil moisture is used, (2) irrigate when 75 percent of the available soil moisture is depleted,c (3.) irrigate when 50 percent of the available soil moisture is depleted and (4) irrigate when only 60 2 5 percent of the available soil moisture is depleted . The results following the various strategies are given in the Appendix. The model estimates consumptive use and irrigation needs and than makes the irrigation. The time period involved is from the date of planting the cotton until harvest is complete, 228 days. The column "Day Growth" identifies the date; 6.30 being June 30. The examples were computed with actual data. In an application, initial inputs would be the soil water holding capacity, historical temperature, "K" factors, the rate of growth of the root system at various crop stages, daylight hours for each day at given location and the available moisture in the projected root zone at planting time. Each day the recorded mean temperature would be entered to calculate the consumptive use. The output of each day would be the percent available moisture remaining in the root zone at the end of five days hence. The decision of whether or not to irrigate would be based upon the percent available moisture figures. This model provides a simple and quick way to ascertain irrigation needs without resorting to field soil sampling. A few interesting aspects of this model should be noted. The model assumes that water is equally available to the plants between field capacity and the wilting point. ITiis equal availability is shown by the daily consumptive use being unrelated and unaffected by the current percent available moisture, A plant is presumed to transpire the same quantity of moisture regardless of whether the soil is very dry or an irrigation has just occurred. This leads to a second observation about the frequency of irrigation and the quantity of water 61 used for the season. The first criterion of allowing the soil water to be depleted before irrigating required" 5 irrigations with a total water use of 44.04 inches. Criterion two of allowing 75 percent of the moisture to the depleted required seven irrigations with a total water use of 44.03 inches. The third criterion of 50 percent required 13 irrigations with 44.05 inches of water used. The very wet treatment of never allowing the available soil moisture to be more than 25 percent depleted required 54 irrigations with a water use of 44.02 inches. The above calculations assume no wastage; only that quantity of water is applied which will exactly bring'the soil to field capacity. With the very wet treatment irrigations were occurring as often as every day. There is some question of the validity of calculating consumptive use for such short periods as one day. But, in order to properly get at the true nature of the response of crops to the application of irrigation water, we must eventually deal with such short periods. It is not unusual to find crops in arid regions using .4 of an inch of water per day during hot periods. Such a usage could deplete the available moisture in two feet of the root zone in 5 or 6 days. The effect of the moisture depletion upon plant growth and more importantly yield, needs to be discovered. A movement away from accepting the "equal facility" theory would be to specify irrigating at some point short of total moisture depletion at 75, 50 or 25 percent for example. Using an intermédiate criterion would result i,n more rapid growth and higher yields generally. 62 The optimal strategy of criteria to use throughout the growing season has not been ascertained and would be partially a subject for economic analyses. The irrigation scheduling model presented is limited also because the consumptive use calculations are unaffected by the stage of moisture depletion. A refinement should be made wherein specification is made of change in consumptive use as total available moisture decreases. Further refinements of this model should prove very worthwhile. Summary The incorporation of a water input into production functions presents some special problems. Traditionally, one would approach the problem by developing a series of experiments in which the quantity of water applied is a treatment. The disadvantage of doing so is of assuming that the distribution of water over time has no effect upon yield. Obviously, this is not the case, as there exists a physical limit to the water storage capacity of any soil and the question of availability of water from the soil for plant growth. Various opposing schools of thought have arisen concerning water availability. One current theory, the "equal facility" theory, states that water is used with equal ease by plants between field capacity and the permanent wilting point. This approach suggests that one should irrigate only when the moisture tension indicated approached the permanent wilting point. Another current theory, the "inverse function" theory, is given the most recognition today. It contends that plants respond differently 63 to soil moisture content and that changes in the moisture regime during plant growth result in corresponding changes in yields of the irrigated crops. According to this school of thought, the growth rate progressively diminishes as the available soil moisture content falls below field capacity, and growth completely ceases when the permanent wilting point is reached. The theory of "inverse function" defends the position that the growth rate of plants is an inverse function of the soil water stress. This stress or tension with which the moisture is held by the soil is a direct function of (1) soil structure, (2) the osmotic effect of salts present in the soil, and (3) the level of depletion of the available soil moisture. In order for water to flow from the soil into the plant tissues, the stress within the plant cells must be greater than in the soil. Accordingly, it is possible to vary the level of yields through variation of water input per unit of land as well as the frequency of irrigation. Hie major problem of the estimation of such functions is their specification and choice of independent variables. Some researchers specify the independent variable in terms of an index called "integrated moisture stress" (I. M. s.). This index is obtained by integrating all the local and instantaneous moisture tension values oiver the relevant soil profile and growth period. Thus it represents the overall weighted soil moisture tension the soil, which determines the degree of 64 availability of soil water to plants. According to Beringer, "the idea of relating plant growt first to moisture tension and only later to water quantity has been a great step forward in agronomic research with water. As a result, a more widely applicable general relationship between moisture conditions and plant growth has been developed. The introduction of the concept of moisture tension has made it possible to construct production functions which are, in a sense, independent of soil type and water quantity."^ A computer model, basically a budget, which keeps account of soil moisture indicates the limitations of accepting the "equal facility" theory. Based upon the empirical formulation for estimating consumptive use, the model indicates constant water use with widely varying irrigation regimes. As far as the actual nature of the water production function is concerned, it may be noted that existing biological theory does not provide precise information beyond the general knowledge that there exists, at least over a certain range, a positive yield response to increasing quantities of water. One may despair that this makes it impossible to obtain an accurate estimate of the production function, since such an estimate requires the knowledge of the parametric form of the function. However, even an approximate estimate is one of great economic importance and is more useful than none at all. Beringer, op. cit., p. 13. 65 THE PRODUCTION FUNCTION FOR CROPS USING IRRIGATION WATER The theoretical analysis of input-output relations in irrigation in terms of a production function have long been presented. It seems, however, that the production function approach has not yet been accepted in its own right. This lack of acceptance and understanding is evident by the statements of scientists working in the irrigation field. The Bureau of Reclamation speaks of "water requirements" for crop production implying fixed inputs per land unit with apparently no allowance for sub- stitutability of factors. Many publications state that projects must be assured of enough water to allow maximum physical product forthcoming from each acre of land. The review by Clark^ on the economics of irri gation in various countries also shows the main approach to be of fixed water inputs per land unit. Requirement projections implicitly assume that the projected amount of the input will be used regardless of supply costs. The increasing scarcity and the associated higher development costs of new water supplies both in the United States and in other arid parts of the world, make it necessary that one know more about the nature of water-yield relationships. Water-yield relationships are important from the viewpoint of efficient farm management, but even more so for water policy where the productivity of water within and between alternative uses will be used increasingly as a criterion in the interpretation and 1 Colin Clark. The economics of irrigation. New York, N.Y., Perga- mon Press. 1967. reformulation of water laws and in the formulation of federal, state, and local government water development projects. Arid regions possess large acreages of fertile soil that become highly productive when irrigation water is applied. Since water is usually not free, it is paramount to efficient irrigation practices to know the behavior of crops on various soils to additional water as well as how to adjust the amount of moisture applied to achieve optimum crop production. The proper form of the water-yield relationship could be gained by ob serving crop response under controlled conditions. Usually, the initial reaction of economists confronted with the problem of determining a water response function in agriculture is to set up a series of experiments in which water quantity is the independent variable and crop output the dependent variable. This approach has been used by economists in most production function studies carried out to date. It appears to be applicable in all those instances in which either the distribution of the input factor over the production period may be fixed on technological grounds or the storage capacity of soils can be assumed to be such that variations in the distribution of nutrients will not materially affect total output, as in the case of most fertilizers. Agronomists have, on the whole, given up the idea of trying to de termine a production function for water simply by applying various quanti ties of water on a number of plots and measuring the resulting production response. The problem facing one using the quantity of water approach is that the involvement of time as a variable through the production cycle 67 "k allows an infinite number of water application patterns- Researchers are now concentrating on finding plant-water relationships which are, in a sense, independent of various soil types. Towards a water production function A significant contribution in clarifying the theoretical concepts concerning the water production function was made by Charles V. Moore^ of the University of California at Davis. A paper presenting the approach was published in the Journal of Farm Economics in November, 1961. Moore, in addition to presenting a model of the production function for water within one irrigation cycle, concludes that economists have misinterpreted the results of irrigation experiments due to a lack of understanding of the basic underlying plant-soil water theory and that they are unable to measure the value of water at more than one point on the production sur face because of using only total annual water inputs in their data. How ever, his presentation is somewhat confusing because of the telegraphic style and lack of detail of the optimizing model. One is left to specu late upon what assumptions were in the author's mind at the time of writing. The purpose here is to build upon the concepts presented by Moore to further refine and improve the usefulness of the approach for estimating *In contemplating construction of a reservoir, more than the static demand schedule should be known. Also involved is the additional amount that farmers would be willing to pay for their supply of water if it were made available at different times of the year. The problem here is not how much water would farmers buy at a given price, but rather, how much would farmers value similar quantities of water at different times of the year. In brief, the problem is whether an acre-foot of water has the same value in July as in May. ^Cbar^es V, Moore. A genergi analytical framework for estimating the production function of crops using irrigation water. Journal of Farm ' Economics 43:876-888. 1961. 68 the production function for crops using irrigation water. Assumptions The most critical assumption now invoked is that the growth rate of plants is an inverse function of the soil water stress. This is critical because the assumption of any other relationship would dictate an entirely different model and likely lead to quite different results. This choice is based upon the careful examination of the main theories and information concerning soil-water relationships now held by agronomists. The soil moisture stress concept appears to be the most plausible and best ap proximation to reality that is currently available. As a first approximation it will be assumed that there is a given volume of soil to wet which remains constant throughout the season. Al though this assumption is not very realistic as the root growth of crops normally continues to contact an increasing amount of soil during the growth cycle, it will not alter the results appreciably. Further, the soil will be considered as a homogeneous mass and it will be assumed that the soil moisture is removed equally from all parts of the soil so that the moisture is always evenly distributed throughout the profile. The only factors allowed to vary are (1) the amount of water applied and (2) the frequency and time of application. The soil is wetted, at irrigation, only to field capacity and no more; therefore, no waste. In addition the wetting cannot be less than field capacity. Some immediate results of the above assumptions are: (1) the amount 69 of water applied each irrigation is related to the frequency of applica tion ; as the frequency increases, the amount of water applied each time decreases : (2) the total water applied per season to a crop is a function of the frequency of application,; as the frequency of application increases, the total amount of water applied increases; (3) the optimal frequency will determine the amount of water applied and the optimal frequency is determined by the relationship between frequency and growth. Nature of the model To be ascertained is the nature of the input-output response and the determination of the optimal frequency which will in turn indicate the op timal amount of water to be applied- The causal relationships are de picted below, the arrows indicating the direction of causation. stress——#- production——^ frequency water ^ growth—* yield function of applied irrigation The relation between growth rate and moisture stress is an inverse one: as the stress increases the growth rate decreases at an increasing rate. The implication then for the production function is that as the stress increases, the total product increases, but at a decreasing rate so that diminishing returns are expected. It should be remembered that a model such as this is not timeless. In fact time as a variable is one of the most important aspects of the model. Envisioned here is a produc tion function where stress or moisture availability is necessarily con nected with time because the passage of time is the only way the moisture stress can be changed. One should envision the production function as 70 beginning at the time of irrigation with the independent variable being the moisture available and the relation of growth to decreasing moisture availability being a function which is increasing but at a decreasing rate as depicted in Figure 15, By definition, the soil is at field capacity just after irrigation at which time the most rapid rate of growth is experienced. Presumably, then, if the soil was kept at field capacity over time, the total growth would in crease at a constant rate and the maximum possible growth would be achieved. A broken line representing a constant growth rate and a solid line repre senting actual growth rate after an irrigation are shown in Figure 15. As soon as the irrigation occurs the decision to be made is whether to irrigate again or to wait another time period. At each subsequent time period, that same decision must be made. The question then is raised as to how long to wait before again applying irrigation water. There are only two courses of action (1) irrigate or (2) do not irrigate and wait another period. If water is applied a new function comes into effect. If no water is applied and the decision is made to wait another period, yield is lost due to not having moisture at an optimum. When an irrigation takes place, a cost is incurred: cost for water and labor of application. A cost is also incurred by not irrigating: the cost of yield lost by waiting another period. A crop production response over the production season may be thought of as being restrained by the earliest planting date and the latest harvest date. For simplicity it will be assumed that planting and harvest are rigid set times. Then within the growing season the problem facing a 71 Constant growth rale "O 0) "ctuai growth rate, o after irrigation 4-> %_ Decreasing available moisture, increasing moisture stress, increasing time .'igure 15- production function for one irrigation cycle (hypothetical) 72 farmer is the optimal application of water over this time period. It is assumed that to achieve any production at all at least one irrigation must be made at which time the soil profile is filled. The application of irrigations may be thought of as a discrete process over time as indicated in Figure 16. In attempting to characterize crop response, in addition to requiring at least one irrigation, the following restrictions will be held: (1) there is a maximum growth rate possible given all other factors are held constant; (2) assuming no wastage, there is a maximum to the amount of water which can be applied; and (3) each irrigation cycle is subject to diminishing re turns. Figure 17 illustrates a possible irrigation regime for a hypotheti cal crop. Time as a variable The major unknown in such a representation as Figure 17 is the timing of the water application. The response of the crop in any irrigation cycle is dependent not only on the availability of moisture to the plant but also the time since the previous irrigation and the crop response in the previous period. The previous period response is also dependent upon what occurred before. An additional complicating factor is introduced when it is realized that most crop yields are forthcoming only at the end of the crop season. It is conceivable that the response in a particular period is also dependent upon activities of a later period. For a much simpler problem where no forward linkage exists, let Tj^, T2,.. -, T^ be the time of irrigations ^1' ^2' *'*' respectively; t^, t2, ., t^ be the time periods between Plant ing Harvest '2 '3 '4 rîgure 16. Discrete water application over time Added Yield Harvest Added Yield T ime Figure 17, Nature of crop response to successive irrigations 75 and , T2 and T^, —, and harvest respectively; and , Y^, , Y^ be the yields of each individual period. Total yield for the crop season is then the sum of the yields in each individual irrigation cycle. It follows that t^ = Tg - T^, ^2 ~ "^3' "^2' ^n " % " ^n' yield in period one (Y^) is a function of t^ and or generally Ti. Vi' • In particular Yj = tjCtj. Tj) Y2 " f 2(^2 ' ^2' ^ = 'A- •'i.- Vl> • The sum of the Y^ equals the total yield for the crop season. As the number of periods being considered increases, the number of feasible irrigation regimes becomes very large. With only twelve time periods, the number of strategies is 2048. The problem is to choose that strategy which is optimal with respect to the objectives of the firm. From the economic standpoint it is desirable to maximize profit from the appli cation of water to the plants. A process or strategy is therefore sought which will optimize crop response to water over time such that net return to the operator is at a maximum. Considering the irrigation season composed of a finite number of time periods presents a problem which is made up of a discrete number of de cisions. At the beginning of each period there exist two alternatives; (1) irrigate, or (2) do not irrigate. If the decision is to irrigate then 76 a new irrigation cycle is begun. A decision not to irrigate means that crop growth follows the same function until the next period. If the decision is made not to irrigate the return for the period is somewhat less than for the previous period. Associated with the diminishing returns is a decreasing marginal revenue. Since the moisture depleted during the period must be replenished at some future time, the marginal cost of irrigation is not zero but for this example is a constant positive amount. If an irrigation is applied, a cost is incurred for water and labor. The response of the crop then follows a new production function cycle. Profit maximizing solution For illustrative purposes data shown in Table 2 have been plotted in Figure 18. Table 2 gives values per period of the cost of irrigation to fill the soil to field capacity; the marginal returns; the total revenue; the profit for the cycle; and the profit per unit of time. Figure 18 indicates the behavior of the cost and revenue functions over time. Conventional economic wisdom would lead one to maximize the return for the irrigation cycle. Such a decision leads to the equating of the marginal cost of irrigation with the marginal revenue. From the table or the figure it is easy to see that marginal revenue equals the one dollar marginal cost at period five. At period five, the profit equals $2.50 (7.50-5.00) which is the maximum for the profit function. Profit for the cycle is also $2.50 at period four so it would have been just as well to Total revenue ^ o O rr igat ion cos t N 0)L. — o Profit for cycle Marginal revenue Marginal cost rof "ft per il. Ar i. ! lime periods Figure 18. Costs and returns for hypothetical crop under irrigation Hypothetical crop production and irrigation cost data :io Cost of irrigation lo Returns Total Profit Profit per ti fill to field capacity per period revenue for cycle unit of time 1 3.00 2.00 2.00 -1.00 -1.00 2 3.00 1.75 3.75 .75 .37 3 3.00 1.50 5.25 2.25 .75 4 4.00 1.25 6.50 2.50 .62 5 5.00 1.00 7.50 2.50 .50 6 6.00 .75 8.25 2.25 .37 00 7 7.00 .50 8.75 1.75 .25 8 8.00 .25 9.00 1.00 .12 9 9.00 .00 9.00 0.00 .00 10 lO'.OO - .25 8'. 7 5 -1.25 11 11.00 - .50 8.25 -2.75 12 12;00 -;75 7.50 -4:50 13 -1,00 6.50 14 -1.25 5.25 15 -1.50 3.75 16 -1,75 2.00 17 -2.00 0.00 "Dead" 79 irrigate one period earlier. The ambiguity enters in because of using discrete data to form continuous functions. If irrigation was applied at period four, the profit for a 100 period cropping season would equal $62.50 (100/4 • 2.50). However, when dealing with time as a variable it is not appropriate to maximize profit per cycle, but rather maximize per unit of time. The profit per unit of time has been plotted in Figure 18. Profit per unit of time is a maximum at period three where profit equals $.75. Using profit per unit of time as a criterion yields a seasonal profit of $75.00 ($.75-100 or 100/3-$2.25). Figure 19 shows the possible strategies that the irrigator faces and the shape the production function may take over time. Any one of the paths are feasible for the operator to follow. In the above example profits will be maximized if irrigation is applied every third period. The dynamic nature of crop response The response of crops to water over time and the decision makirg process is really a dynamic programming problem if factors are allowed to vary to approximate reality. A problem as simple as the one just described can be solved by hand because it was assumed crop response was identical and measurable in each period; irrigation costs increased at a constant rate; and each period was independent of other periods so that the period yields were additive. In a real world problem one would expect crop response to vary dramatically in relation to the stage of crop growth. In addition interdependency would exist between periods. The response in period eight I ime Figure 19. hypothetical yield alternatives with varying Irrigation strategies 81 might be affected by action taken in period four. The response in period 26 might be affected by the action or response of period eight. The in- terdependency may well work both ways. The environment of the plant may change the response function from day to day. There may be extremely "critical" periods wherein the response to water is many times greater than at any other time. Appropriate analysis would take into account all factors and then indicate what course of action would be optimum. Un fortunately, the relevant coefficients for such analyses are not yet known but as progress is made in ascertaining them, application can start in a crude sort of way using recursive dynamic programming. Because data is not available for estimating the parameters of the true response functions of crops to irrigation water no such analysis can be made here. Results of an experiment which goes a long way toward providing insight into the effect of additional water upon plant growth will be discussed in light of the paucity of more advanced experimental information. Summary The production function approach for crops using irrigation water has yet to be accepted in its own right. Scientists still speak of "water requirements" and "water duty" implying fixed inputs of water per acre. Part of the lack of acceptance is because of the paucity of data suitable for production function analysis. Researchers have generally given up the idea of a production function based on quantities of water applied for the total crop season. Recognition is being given the more important factor of time of water application. Pioneering the way for the change 82 in approach to crop response studies were Moore and Beringer. Most crops exhibit diminishing returns to more frequent applications of water. An important underlying determinant of crop response is the soil moisture stress. The concept of soil moisture stress provides a basis of a production function which is somewhat independent of soil types and other exogenous influences. The true response of crops to irrigation water is not known with any degree of certainty. A production response for the season is made up of many sub-period responses which may be quite similar or very different de pending upon the crop under consideration. Unfortunately, the interaction between irrigation cycles within the crop season are not well understood. The irrigation problem is made up of a large number of decisions over time, all of which are not independent. The value of economic application to the irrigation production func tion is to assist in maximizing the profit due to irrigation. The simple model presented, assumed a large degree of independence between periods and indicated maximization of profit per unit of time as the appropriate criterion for maximizing the profit to water application. The characterization, of the production function for specific crops is a dynamic programming problem in which an optimal strategy would be ascertained given the information about the response of the crop in various sub-periods. Specification might be made as to critical stages of plant development which would be accounted for by such a program. The problem is made up of a large, yet discrete, number of sub-problems which need to be solved to obtain the optimal solution for the crop season. 83 The economic optimum as given in the previous example could be related quite precisely to the soil moisture tension for each stage of plant de velopment. The empirical requirement is to specify those moisture tensions which will be optimum in the sense of maximizing profit to the water application. In addition to the determination of the optimal application of an unrestricted quantity of water, the more usual condition of limited water supply could be invoked and the profit maximizing application of various limited supplies could be ascertained. Differences in returns forthcoming from alternative strategies would indicate the value of supplemental water supplies. 84 AN EMPIRICAL PRODUCTION FUNCTION FOR CORN USING IRRIGATION WATER AND NITROGEN FERTILIZER In the previous section attention was given to the specification of the conceptual framework of the response of crops to the application of irrigation water. The true response of any crop to water application is a dynamic problem with time as one of the most important factors affecting the response. No experiments have yet been conducted or designed which would provide enough data for the estimation of the parameters of the dynamic water response. Indeed, such experiments would prove very ex pensive and time consuming so it is not surprising that dynamic response functions have not been estimated. In fact very few static water response functions are available for analysis. Of the static functions that are available, most are estimates using total quantity of irrigation water ap plied during the season. Because data is not available for estimating the parameters of the true response functions for crops using irrigation water, no such analysis can be made at the present. However, the parameters of a quadratic equation were estimated for corn using irrigation water and nitrogen fer tilizer from the results of an experiment designed for the purpose of pro duction function estimation. The estimated production functions relate the grain and forage production of corn to the inputs, water and nitrogen. Although the functions estimated are not dynamic as characterized in the previous section, some of the functions include elements of time by the nature of the irrigation treatments. Lacking data to achieve the ideal, 85 the second best will be investigated. The results of the corn experiment provides considerable insight into the effect of additional water upon plant growth. The nature of crop response given by this experiment will be discussed because of the paucity of more advanced experimental informa tion. Framework of the corn experiment A field experiment was conducted on corn at Colorado State University during the 1968 crop season. The study was established on Nunn clay loam (formerly Fort Collins clay loam) soil at the Colorado Agronomy Research Center. The design of the experiment, which was intended for the estimation of a production surface, was an incomplete factorial with five levels of nitrogen fertilizer and five irrigation regimes. Each of the three blocks contained 22 plots. The combination of factors used in input space is given in Figure 20. Within a block, each of the treatments designated by x*s were replicated twice and each of the o treatments were replicated once. Thus the 13 treatments provide 22 plots per block. The nitrogen fertilizer treatments ranged from 0 to 200 pounds per acre in increments of 50 pounds. The water treatments were in atmospheres of tension and as shown were 0.7, 1.0, 3.0, 6.0 and 9.0. The water treatments consisted of maintaining the root zone soil moisture tension equal to or less than the level indicated. As the plants used moisture the tension increased from field capacity. Whenever the recorded tension in a particular plot 86 200J 150- 100. 50- 0_ i 1 Î 1 ; 0.7 1.0 3.0 6.0 9.0 Maximum soil moisture tension in atmospheres Figure 20. Experimental design showing combinations of factors which comprise the treatments 87 reached the specified treatment, an irrigation was applied sufficient to bring the root zone to field capacity once again. Therefore the amount of irrigation water applied during the season was a result of each treat ment and not a treatment per se. At planting time the soil profile was at field capacity. It should also be noted that each plot was separate and that it was not a split-plot design. This design allows the examination of block by treatment interactions. Most of the points are concentrated about the periphery of the factor space where they are most useful for estimating the parameters of a quadratic function. The plot size consisted of eight rows of corn with 28 inch spacing and 35 foot lengths. The corn was planted on May 9, 1968 with a 105 day hybrid (Kitely K4-17). The plots were harvested September 16-20 for corn silage yields and when the corn matured, grain yields were taken. The ac tual plot yields of grain and forage are given in Tables 3 and 4. Grain yields were adjusted to 15.5 percent moisture and forage yields are given in oven dry weights. The soil profile contained an average of 5.56 inches of available moisture at planting time. Rainfall received on the plots during the season totaled 5.9 inches. Thus each plot had 11.46 inches of moisture available in addition to the irrigation applications. Production function estimation The data was analyzed using multiple regression techniques and fitting the quadratic equation to the data of the form 88 Table 3. Plot grain yields on pounds per acre basis adjusted to 15.5 percent moisture 5502.1 8080.3 7714.9 I 5616.4 8491.8 9005.8 5131.7 8570.7 8277.5 200 II 5802.8 9015.0 9306.0 4828.2 6529.1 6392.3 Q) III U 7563-0 O 5433.7 6970.8 cd u o 6549.8 7432.2 I Oh 0) 150 6599.9 8507.8 II u 4-) 6174.0 7031.8 III •H (W o 6926.0 7237.1 7895.4 I m 5770.7 7163.9 7144.9 'O c D o 5741.1 6655.1 8540.1 CO A 100 II 6228.7 8035.3 7585.2 OG f—ICQ 4-)CO 3923.8 7432.1 8147.4 c III i 5487.9 6701.9 7847.2 0) W 4856.1 8350.3 I W Q) N 50 6323.1 8447.6 II •H r—' U 5615.8 8446.9 III - d) 54 4918.6 4506.4 6117-6 I 5559.7 6761 .0 7087.5 7548.0 7791.3 0 5469.8 II 42 94.1 5630.3 6931-9 5568.1 7052.0 7474.8 III 2137.7 6014.6 82 09.1 9.0 6.0 3.0 1.0 0.7 Irrigation treatments - Atmospheres of tension 89 Table 4. Plot forage yields on pounds per acre basis oven dry weights (Multiply by 4 to get fresh weight) 8,698 9,556 8,847 9,967 12,916 12,020 8,586 9,482 13,177 8,623 12,244 13,103 7,765 11,759 12,692 8,138 10,639 12,543 9,631 14.484 10,826 12,282 9,556 8,138 9,370 10,602 11,572 9,594 12,095 10,900 9,706 11,684 12,394 9,519 11,236 11,236 8,586 12,319 14,596 7,167 10,527 13,737 7,018 12,618 10,378 13,252 8,922 11,535 7,391 6,831 9,967 8,325 10,900 11,199 9,258 10,452 11,236 7,839 9,407 11,012 6,383 10,826 12,132 4,218 9,071 12,282 9.0 6.0 3.0 1.0 0.7 Irrigation treatments - atmospheres of tension 90 Y = bo + Xi + b2 ^2 " ^11 " ^22 ^2 ^12 *2 where: Y = yield of corn grain or forage = water X2 = nitrogen fertilizer b^ = coefficients to be estimated The yields of both grain and forage are recorded in pounds per acre. The water input was characterized by atmospheres of tension of the soil moisture. Nitrogen fertilizer was applied in pounds per acre. The estimating procedure used was multiple regression. The values of and X2 were coded when used to estimate the coefficients to pro vide orthogonality of the treatments. The procedure used to code the treatment was to calculate the mean of the treatment values and then subtract the mean from the actual treatment value. For example, the mean of the moisture tension treatments (0.7, 1.0, 3.0, 6.0, and 9.0) was 3.94. Thus the coded values corresponding were -3.24, -2.94, -.94, 2.06. and 5.06. The values used to correspond to the nitrogen treatments were -2, -1, 0, 1, and 2. Table 5 gives the conversion from actual to coded values. The analysis was run using moisture tension as a variable and then using inches of water applied as a variable. The results of both models will be given. A comparison showing the relationship between soil moisture stress and inches of water applied is given in Table 6 and Figure 21. 91 Table 5. Conversion table from coded X values to actual X values for the production function based upon moisture stress XÏ Pounds X2 Atmospheres Coded nitrogen Coded stress value fertilizer value 0.7 -3.24 0 -2.0 1.1 -2.84 10 -1.8 1.5 -2.44 20 -1.6 1.9 -2.04 30 -1.4 2.3 -1.64 40 -1.2 2.7 -1.24 50 -1.0 3.1 - .84 60 - .8 3.5 - .44 70 - .6 3.9 - .04 80 - .4 4.3 .36 90 - .2 4.7 .76 100 .0 5.1 1.16 110 .2 5.5 1.56 120 .4 5.9 1.96 130 .6 6.3 2.36 140 .8 6.7 2.76 150 1.0 7.1 3.16 160 1.2 7.5 3.56 170 1.4 7.9 3.96 180 1.6 8.3 4.36 190 1.8 8.7 4.76 200 2.0 9.1 5.16 Table 6. Relationship between the soil moisture stress and the seasonal inches of irrigation water applied Atmospheres Inches of stress water applied 9.0 0 0 6.0 2.76 3.0 5.15 1.0 5.88 0.7 8.83 92 8 7 en a) k 0) a oen e c•M 2- ¥ -i ' ' - mtm 34567 10 Inches of water applied Figure 21. Graph showing relationship between moisture stress and water applied. 93 Moisture tension and nitrogen as variables The analysis of variance for both grain and forage using moisture tension and nitrogen as variables is given in Tables 7 and 8. The lack-of-fit is not significant in either grain or forage. The blocks X treatments effect is just significant at the 5% level for forage. The significance of the blocks X treatments indicates a degree of soil variation affecting the forage response. All sources of error: lack-of-fit, block X treatments, and within blocks were pooled for both grain and forage. The ANOVA for both grain and forage with the error pooled is given in Tables 9 and 10. Table 7. Grain ANOVA using moisture tension Source of variation Degrees of Sum of Mean F-ratio freedom squares square Blocks 2 5,425,630 2,712,815 Treatments 12 89,226,491 7,438,874 Regression 5 83,697,226 16,739,445 Lack-of-fit 7 5,529,265 789,895 1.096 Blocks X treatments 24 13,970,105 582,087 .807 Within blocks . 27 19,461,624 720,801 Total 65 128,083,850 94 Table 8. Forage ANOVA using moisture tension Source of variation Degrees of Sum of Mean F-ratio freedom squares square Blocks 2 5,076,628 2,538,314 Treatments 12 175,340,730 14,611,727 Regression 5 170,065,647 34,013,129 Lack-of-fit 7 5,275,083 753,583 .564 Blocks X treatments 24 63,582,987 2,649,291 1.98 Within blocks 27 36,090,857 1,336,698 Total 65 280,091,202 Table 9. Grain ANOVA with error pooled using moisture tension Source of variation Degrees of Sum of Mean F-ratio freedom squares square Blocks 2 5,425,630 2,712,815 Regression 5 83,697,226 16,739,445 24.92** Error 58 38,960,994 671,741 Total 65 12 8,083,850 = .654 95 Table 10. Forage ANOVA with error pooled using moisture tension Source of variation Degrees of Sum of Mean F-ratio freedom squares square Blocks 2 5,076,628 2,538,314 Regression 5 170,065,647 34,013,129 18.8** Error 58 104,948,927 1,809,464 Total 65 280,091,202 = .607 We then get the following estimates of the coefficients of a quadratic function as given in Tables 11 and 12. Levels of significance are indicated by (**) for 1%, (*) for 5%, and (+) for 10%. Table 11. Coefficients of grain production function using moisture tension Coefficient Standard error b^ -- intercept 7 043 224.7 b]^ — irrigation -302.3** 42.5 b2 — nitrogen 251.1** 87.0 2 b^^ — (irrigation) -4.87 15.0 2 b22 — (nitrogen) -76.51 55.6 b22 — interaction I x N 1.64 24.7 96 Table 12. Coefficients of forage production function using moisture tension Coeff icient Standard error \ - intercept 10,728 368.8 irrigation -483.8** 69.7 h " ^>2 "*— nitrogen 258.3+ 142.8 2 - (irrigation) 22.47 24.6 ^1- 2 - (nitrogen) -222.50* ^22 91.2 bi2 —- interaction I x N 13.74 40.6 About 65 percent of the grain yield variation is explained and about 61 percent of the forage yield variation is explained by the above analyses. The water-nitrogen interaction is negligible and there is almost no curvature in the response to moisture. What non-linear response exists, is concave upward for forage and concave downward for grain. The nitrogen effect is quite strongly curved concave downward for both grain and forage. Inches of irrigation water applied and nitrogen as variables Inches of irrigation water applied and nitrogen fertilizer were then tried as variables in the equations. The nitrogen treatments were coded as before. The inches of water applied were 0, 2.76, 5.15, 5.88, and 8.83. The corresponding coded values used were -4.52, -1.76, 0.63, 1.36, and 4.31. Table 13 gives the conversion from actual to coded values. 97 Table 13. Conversion table from coded X values to actual X values for the production function based upon inches of water applied Xl X2 Inches of water Coded Pounds nitrogen Coded applied value fertilizer value 0 -4.52 0 -2.0 .5 -4.02 10 -1.8 1.0 -3.52 20 -1.6 1.5 -3.02 30 -1.4 2.0 -2.52 40 -1.2 2.5 -2.02 50 -1.0 3.0 -1.52 60 - .8 3.5 -1.02 70 - .6 U.O - .52 80 - .4 4.5 - .02 90 - .2 5.0 .48 100 .0 5.5 .98 110 .2 6.0 1.48 120 .4 6.5 1.98 130 .6 7.0 2.48 140 .8 7.5 2.98 150 1.0 8.0 3.48 160 1.2 8.5 3.98 170 1.4 9.0 4.48 180 1.6 190 1.8 2 00 2.0 The analysis of variance for both grain and forage are shown in Tables 14 and 15. The lack-of-fit is not s ignificant in either case. The effect of blocks X treatments is also non-significant for both grain and forage. Once again all sources of error: lack-of-fit, block x treat ments, and within blocks were pooled for both grain and forage. The reduced ANOVA is given in Tables 16 and 17 showing the pooled error term. 98 Table 14. Grain ANOVA using inches of water applied Source of variation Degrees of Sum of Mean F-ratio freedom squares square Blocks 2 5,425,630 2,712,815 Treatments 12 89,225,492 7,435,541 Regression 5 81,270,146 16,254,029 Lack-of-fit 7 7,956,346 1,136,621 1.626 Blocks X treatments 24 14,560,179 606,674 .868 Within blocks 27 18,871,548 698,946 Total 65 128,083,849 Table 15. Forage ANOVA using inches of water applied Source of variation Degrees of Sum of Mean F-ratio freedom squares square Blocks 2 5,076,628 2,538,314 Treatments 12 175,340,730 14,611,727 Regression 5 165,801,254 33,160,251 Lack-of-fit 7 9,539,476 1,362,782 .956 Blocks X treatments 24 61,165,970 2,548,582 1.79 Within blocks 27 38,507,874 1,426,218 Total 65 280,091,202 99 Table 16. Grain ANOVA with error pooled using inches of water applied Source of variation Degrees of Sum of Mean F-ratio freedom squares square Blocks 2 5,425,630 2,712,815 Regression 5 81,270,146 16,254,029 22.78** Error 58 41,388,073 713,587 Total 65 128,083,849 = .635 Table 17. Forage ANOVA with error pooled using inches of water applied Source of variation Degrees of Sum of Mean F-ratio freedom squares square Blocks 2 5,076,628 2,538,314 Regression 5 165,801,254 33,160,251 17.61** Error 58 109,213,320 1,882,988 Total 65 280,091,202 = .592 Tables 18 and 19 give the estimates of the coefficients of the quadratic function. Levels of significance are indicated by (**) for 1%, (*) for 5%, and (+) for 10%. The nitrogen-water interaction is negligible and there is very little curvature in the response to moisture for forage but a rather 100 Table 18. Coefficients of grain production function using inches of water applied Coefficient Standard error ^ " intercept 7258 211.1 irrigation 284.66** 32.87 ^2 nitrogen 246.91** 70.75 2 b^l -- (irrigation) -29.96** 12.2 2 ^22 *"- (nitrogen) -78.57 60.45 bi2 --• interaction I x N 3.13 20.16 Table 19. Coefficients of forage production function using inches of water applied Coefficient Standard error intercept 11,077 329.8 — irrigation 426.34** 51.35 h -- b2 - nitrogen 2 92.41* 11.05 (irrigation)^ -15.27 19.0 ^11 -- 1^22 """" (nitrogen)^ -237.81* 94.46 b-, 9 — interaction I x N -20.80 31.50 101 significant non-linear moisture response in grain. The effect of the non-linear response is just reversed for nitrogen. About 63 percent of the variation in grain yield and about 59 percent of the variation in forage yield is explained by the above analyses. Yield predictions Using the estimated production functions the yields of grain and forage can be estimated for varying inputs of nitrogen and water. Tables of predicted yields were constructed for all four fitted equations. Resulting yields are given in pounds and bushels per acre for grain and in pounds and tons per acre for forage. Thus eight tables of predicted yields are given for comparative purposes- See Tables 20 through 27. The yields were estimated by using the following four equations. For moisture tension as a variable Ygrain = 7 043 - 302.+ 251.1%% - 4.87X.2 _ yô.SlXg^ + l.G^X^Xg and 9 2 Yforage = 10,728 - 483.SX^ + 258.3X2 + 22.47Xj^ - 222.5X2 13.74X^X2 where X^ is the coded specified moisture tension and X2 is the coded specified nitrogen level per acre. For inches of irrigation water applied as a variable Ygrain = 7258 + 284.66Xj^ + 246.91X2 " 29.96X^^ - 78.57X2^ + 3.13X^X2 and Yforage = 11,077 + 426.34Xi + 292.41X7 - 15.22Xj^ - 237.81X2^ - 20.8X^X2 where X^ is the coded specified inches of water applied and X2 is the coded specified nitrogen level per acre. Table 20. Predicted yields of com grain in po'onds per acre for selected nitrogen levels and soil moisture tensions Soil. XoistTire Tension in Atmospheres at which '.-Jater is Applied 9.1 8.7 8.3 7.9 7.5 7.1 6.7 6.3 5.9 5.5 5.1 200 5567 5705 5843 5979 6113 6246 6377 6506 6634 6761 6886 190 5573 5712 5849 5935 6120 6253 6384 6513 6642 6768 6893 180 5573 5712 5850 5936 6120 6253 6385 6515 6643 6770 6895 170 5567 5706 5844 5980 6115 6248 6379 6509 ^638 6765 6890 2 160 5555 5694 5832 5968 6103 6236 6369 6496 6627 6754 6879 O 150 5537 5676 5814 5951 6O86 6219 6351 6481 66IO 6737 6862 O t 140 5512 5652 5790 5927 6062 6195 6327 6457 6586 6714 6839 o 2] 130 5482 5622 5760 5897 6032 6165 6297 6428 6557 6684 6810 & 120 5445 5585 5723 5860 5996 6129 6262 6392 6521 6649 6775 § 110 5402 5542 56SI 5818 5953 6087 6220 6350 6480 6607 6733 100 5353 5494 5632 5770 5905 6039 6172 6302 6432 6560 6686 rrj 90 5298 5439 5578 5715 5851 5985 6117 6248 6378 6506 6632 o 80 5237 5378 5517 5654 5790 5924 6057 6I88 63I8 6446 6572 o 70 5170 5311 5450 5537 5723 5858 5991 6122 6252 6380 6506 60 5097 6238 5377 5514 5651 5785 5918 6050 6179 63O8 6434 50 5017 5158 5298 5435 5572 5706 5839 5971 6101 6229 6356 40 4932 5073 5212 5350 5487 5621 5755 5886 6OI6 614^ 6272 30 4840 4981 5121 5259 5395 5530 5664 5796 5926 6054 6l82 20 4742 4884 5023 5162 5298 5433 5567 5699 5829 5958 6085 10 4638 4780 4910 5058 5195 5330 5464 5596 5726 5855 5982 ' 0 4528 4670 4810 4948 5O85 5220 5354 5486 5617 5746 5874 103 Soil Koistiire Tension in Atmospheres at which Water is Applied 4.7 4.3 3.9 3.5 3.1 2.7 2.3 1.9 1.5 1.1 .7 7009 7131 7251 7370 7487 7602 7716 7829 7940 8049 8157 190 7017 7139 7259 7378 7495 7611 7725 7837 7946 8058 8I66 180 7018 7140 7261 7380 7497 7613 7727 7840 7951 8061 8I69 170 7014 7136 7257 7376 7493 7609 7723 7836 7948 8057 8165 160 7003 7125 7246 7365 7483 7599 7714 7827 7938 8048 8156 150 6986 7109 7230 7349 7467 7583 7698 7811 7922 8032 8141 140 6963 7086 7207 7326 7444 7561 7675 7789 7900 8010 8119 130 6934 7057 7178 7298 7416 7532 7647 7761 7872 7983 8091 120 6899 7022 7143 7263 7381 7498 7613 7726 7838 7949 8057 110 6858 6980 7102 7222 7340 7457 7572 7686 7798 7908 8017 100 6810 6934 7055 7175 7293 7410 7526 7639 7752 7862 7971 90 6757 6880 7002 7122 7240 7357 7473 7587 7699 7810 7919 80 6697 6821 6942 7063 7181 7298 7414 7528 7641 7751 7861 70 6631 6755 6877 6997 7116 7233 7349 7463 7576 7687 7796 60 6560 6683 6805 6926 7045 7162 7278 7392 7505 7616 7726 50 6482 6605 6728 6848 6967 7085 7201 7315 7428 7539 7649 40 6397 6521 6644 6764 6384 7001 7117 7232 7345 7456 7566 30 6307 6431 6554 6675 6794 6912 7028 714-3 7256 7367 7477 20 6211 6335 6458 6579 6698 6816 6932 7047 7l60 7272 7382 10 6108 6233 6355 6476 6596 6714 6831 6946 7059 7171 7281 0 6000 6124 6247 6368 6488 6606 6723 6838 6951 7063 7174- Table 21. Predicted yields of corn forage in pounds per acre for selected nitro~en levels and soil moisture tensions Soil i:•oisture Tension in Atmospheres at which Water is Applied 9.1 8.7 8.3 7.9 7.5 7.1 6.7 6.3 5.9 5.5 5.1 200 8598 8692 8792 8900 9015 9137 9266 9403 9547 9697 9856 190 8702 8796 8898 9006 9123 9246 9376 9514 9659 9811 9970 180 8787 8883 8985 9095 9212 9337 9468 9607 9753 9906 10066 170 8855 8951 9055 9166 9284 9410 9542 9682 9829 9983 10145 o 160 8904 9002 9107 9219 9339 9465 9599 9740 9888 10043 10206 <5;O U 150 9035 8937 9141 9255 9375 9503 9638 9780 9929 10085 10249 SX0 9272 9802 10109 10274 ÎH Soil Moisture Tension in Atmospheres at 'vdiich Water is Applied 4.7 4.3 3.9 3.5 3.1 2.7 2.3 1.9 1.5 1.1 .7 200 10021 10193 10373 10560 10754 10955 11163 11379 11602 11832 12069 190 10136 10310 10490 10678 10874 11076 11285 11502 11726 11957 12195 180 10234 10408 10590 10779 10975 11179 11389 11607 11832 12064 12304 170 10313 10489 10672 10862 11060 11264 11476 11695 11921 12154 12395 20 9374 9566 9765 9972 10186 10407 10635 10870 11113 11363 11620 10 9169 9362 9563 9770 9985 10207 10437 10673 10917 III68 11426 0 8946 9140 9342 9551 9767 9990 10220 10458 10703 10955 11214 Table 22. Predicted yields of com %rain in bushels per acre for selected nitroeen levels and soil moisture tensions Soil Moisture Tension in Atmospheres at which Water is Applied 9.1 8.7 8.3 7.9 7.5 7.1 6.7 6.3 5.9 5.5 5.1 200 99.4 101.9 104.3 106.8 109.2 111.5 113.9 116.2 118.5 120.7 123.0 190 99.5 102.0 104.5 106.9 109.3 111.7 114.0 116.3 118.6 120.9 123.1 180 99.5 102.0 104.5 106.9 109.3 111.7 114.0 116.3 II8.6 120.9 123.1 170 99.4 101.9 104.4 106.8 109.2 111.6 113.9 116.2 118.5 120.8 123.O 2 160 99.2 101.7 104.1 106.6 109.0 111.4 113.7 116.0 118.3 120.6 122.8 O 150 98.9 101.4 103.8 106.3 108.7 111.1 113.4 115.7 118.0 120.3 122.5 c a l-'iO 98.4 100.9 103.4 105.8 108.2 110.6 113.0 115.3 117.6 119.9 122.1 o tc 130 97.9 100.4 102.9 105.3 107.7 110.1 112.5 114.8 117.1 119.4 121.6 -p i 120 97.2 99.7 102.2 104.7 107.1 109.5 111.8 114.2 II6.5 118.7 121.0 § 110 96.5 99.0 101.5 103.9 106.3 108.7 111.1 113.4 115.7 118.0 120.2 O I 100 95.6 98.1 100.6 103.0 105.5 107.8 110.2 112.5 114.9 117.1 119.4 •ri Cm 90 94.6 97.1 99.6 102.1 104.5 106.9 109.2 111.6 113.9 116.2 118.4 O I 80 93.5 96.0 98.5 101.0 IO3.4 105.8 108.2 110.5 112.8 115.1 117.4 cÇ 70 92.3 94.8 97.3 99.8 102.2 104.6 107.0 109.3 111.6 113.9 116.2 60 91.0 93.5 96.0 98.5 100.9 103.3 105.7 108.0 110.4 112.6 114.9 50 89.6 92.1 94.6 97.1 99.5 101.9 104.3 106.6 109.0 111.2 113.5 40 88.1 90.6 93.1 95.5 98.0 100.4 102.8 105.I 107.4 109.7 112.0 30 86.4 89.0 91.4 93.9 96.4 98.8 101.1 103.5 10-5.8 108.1 110.4 20 84.7 87.2 89.7 92.2 94.6 97.0 99.4 101.8 104.1 106.4 108.7 10 82.8 85.4 87.9 90.3 92.8 95.2 97.6 99.9 102.3 104.6 IO6.8 0 80.9 83.4 85.9 88.4 90.8 93.2 95.6 98.0 100.3 102.6 104.9 107 Soil Moisture Tension in Atmospheres at vhich Water is Applied 4.7 4.3 3.9 3.5 3.1 2.7 2.3 1.9 1.5 1.1 .7 200 125.2 127.3 129.5 131.6 133.7 135.8 137.8 139.8 141.8 143.7 145.7 190 125.3 127.5 129.6 131.8 133.8 135.9 137.9 140.0 141.9 143.9 145.8 180 125.3 127.5 129.7 131.8 133.9 136.0 138.O 140.0 142.0 143.9 145.9 170 125.3 127.4 129,6 131.7 133.8 135.9 137.9 139.9 141.9 143.9 145.8 160 125.1 127.2 129.4 131.5 133.6 135.7 137.7 139.8 141.8 143.7 145.6 0 1 150 124.8 126.9 129.1 131.2 133.3 135.4 137.5 139.5 141.5 143.4 145.4 I 140 124.4 126.5 128.7 130.8 132.9 135.0 137.1 139.1 141.1 143.0 145.0 I 130 123.8 126.0 128.2 130.3 132.4 134.5 136.6 138.6 140.6 142.6 144.5 120 123.2 125.4 127.6 129.7 131.8 133.9 135.9 138.0 140.0 141.9 143.9 ^ no 122.5 124.7 126.8 129.0 131.1 133.2 135.2 137.3 139.3 141.2 143.2 § ^ 100 121.6 123.8 126.0 128.1 130.2 132.3 134.4 136.4 138.4 140.4 142.4 •p g 90 120.7 122.9 125.0 127.2 129.3 131.4 133.4 135.5 137.5 139.5 141.4 0 80 119.6 121.8 124.0 126.1 128.2 130.3 132.4 134.4 136.4 138.4 140.4 M 1 70 118.4 120.6 122.8 125.0 127.1 129.2 131.2 133.3 135.3 137.3 139.2 tS 60 117.1 119.3 121.5 123.7 125.8 127.9 130.0 132.0 134.0 136.0 138.0 50 115.7 118.0 120.1 122.3 124.4 126.5 128.6 13O.6 132.6 134.6 136.6 40 114.2 116.5 118.6 120.8 122.9 125.0 127.1 129.1 131.2 133.2 135.1 30 112.6 114.8 117.0 119.2 121.3 123.4 125.5 127.6 129.6 13I.6 133.5 20 110.9 113.1 115.3 117.5 119.6 121.7 123.8 125.8 127.9 129.9 131.8 10 109.1 111.3 113.5 115.7 117.8 119.9 122.0 124.0 126.1 128.1 I3O.O 0 107.1 109.4 111.6 113.7 115.9 118.0 120.2 122.1 124.1 126.1 128.1 Table 23. Predicted jrields of corn forage in tons per acre for selected nitrogen levels and soil moisture tensions Soil Moisture Tension in Atmospheres at which iJater is Applied 9.1 8.7 8.3 7.9 7.5 7.1 6.7 6.3 5.9 5.5 5.1 200 17.20 17.38 17.58 17.80 18.03 18.27 18.53 18.81 19.09 19.39 19.71 190 17.40 17.59 17.80 18.01 18.25 18.49 18.75 19.03 19.32 19.62 19.94 180 17.57 17.77 17.97 18.19 18.42 18.67 18.94 19.21 19.51 19.81 20.13 170 17.71 17.90 18.11 18.33 18.57 18.82 19.08 19.36 19.66 19.97 20.29 u 160 17.81 18.00 18.21 18.44 18.68 18.93 19.20 19.48 19.78 20.09 20.41 % ISO 17.87 18.07 18.28 18.51 18.75 19.01 19.26 19.56 19.86 20.17 20.50 O u ]40 17.90 18.10 18.32 18.54 18.79 19.05 19.32 19.60 19.90 20.22 20.55 o H 130 17.89 18.10 18.31 18.54 18.79 19.05 19.32 19.61 19.91 20.23 20.56 ? è 120 17.85 18.06 18.27 18.51 18.76 19.02 19.29 19.58 19.89 20.21 20.54 o 110 17.77 17.98 18.20 18.^ 18.69 18.95 19.23 19.52 19.83 20.15 20.i^9 M ^ 100 17.66 17.87 18.09 18.33 18.58 18.85 19.13 19.42 19.73 20.06 20.39 «M 90 17.51 17.72 17.95 18.19 18.44 18.71 18.99 19.29 19.60 19.93 20.27 o ^ 80 17.33 17.54 17.77 18.01 18.26 18.53 18.82 19.12 19.43 19.76 20.10 (S 70 17.10 17.32 17.55 17.79 18.05 18.32 18.61 18.91 19.23 I9.56 19.90 60 16.85 17.07 17.30 17.54 17.80 18.08 18.37 18.6? 18.99 19.32 19.67 50 16.56 16.78 17.01 17.26 17.52 17.80 18.09 18.44 18.72 19.05 19.40 40 16.23 16.45 16.69 16,94 17.20 17. 17.7^ 18.08 15.41 18.74 19.10 30 15.87 16.09 16.33 16.58 16.85 17.13 17.43 17.74 18.06 18.40 18.75 20 15.47 15.69 15.93 16.19 16.46 16.74 17.04 17.35 17.68 18.02 18.38 10 15.03 15.26 15.50 15.76 16.03 16.32 16.62 16.93 17.26 17.61 17.96 0 14.56' 14.79 15.04 15.30 15.57 15.86 I6.I6 16.48 16.81 17.16 17.52 109 Soil I'oistiire Tension in Atmospheres at t-rfiich Water is Applied 4.7 4.3 3.9 3.5 3.1 2.7 2.3 1.9 1.5 1.1 .7 20.04 20.39 20.75 21.12 21.51 21.9. 22.33 22.76 23.20 23.66 24.14 20.27 20.62 20.98 21.36 21.75 22.15 22.57 23.00 23.45 23.91 24.39 20.47 20.82 21.18 21.56 21.95 22.36 22.78 23.21 23.66 24.13 24.61 170 20.63 20.98 21,34 21.72 22.12 22.53 22.95 23.39 23.84 24.31 24.79 16 0 20.75 21.10 21.47 21.86 22.25 22.66 23.09 23.53 23.98 24.45 24.94 150 20.84 21.19 21.57 21.95 22.35 22.76 23.19 23.63 24.09 24.56 25.05 140 20.89 21.25 21.62 22.01 22.41 22.83 23.26 23.70 24.16 24.63 25.12 130 20.91 21.27 21.64 22.03 22.44 22.85 23.29 23.73 24.19 24.67 25.16 120 20.89 21.25 21.63 22.02 22.43 22.85 23.28 23.73 24.19 24.67 25.16 110 20.84 21.20 21.58 21.97 22.38 22.00 23.24 23.69 24.16 24.64 25.13 100 20.75 21.11 21.49 21,89 22.30 22.72 23.16 23.62 24.08 24.57 25.06 90 20.62 20.99 21.37 21.77 22.18 22.61 23.05 23.51 23.98 24.46 24.96 80 20.46 20.83 21.22 21.62 22.03 22.46 22.90 23.36 23.83 24.32 24^82 70 20.26 20.64 21.03 21.43 21.84 22.28 22.72 23.18 23.65 24.14 24.65 60 20.03 20.41 20.80 21.20 21.62 22.05 22.50 22.96 23.44 23.93 24.44 50 19.76 20.14 20.53 20.94 21.36 21.80 22.25 22.71 23.I9 23.68 24.19 40 19.46 19.84 20.24 20.64 21.07 21.51 21.96 22.42 22.90 23.40 23.91 30 19.12 19.50 19.90 20.31 20.74 21.18 21.63 22.10 22.58 23.08 23.59 20 18.75 19.13 19.53 19.94 20.37 20.81 21.27 21.74 22.23 22.73 23,24 10 18.34 18.72 19.13 19.54 19.97 20.41 20.87 21.35 21.83 22.34 22.85 0 17.89 18.28 18.68 19.10 19.53 19.98 20.44 20.92 21.41 21.91 22.43 Taille 24. Predicted yields of corn %rain in pounds per acre for selected nitrogen levels and irrigation ifater applications Inches of Water Applied (11,46 inches of rain and soil moisture) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 200 5511 5784 6042 6286 6514 6728 6926 7110 7278 7432 190 5524 5797 6055 6298 6526 6739 6937 7121 7289 7442 180 5531 5804 6061 6304 6532 6745 6942 7125 7293 7446 170 5531 5804 6061 6304 6531 6744 6941 7124 7291 7444 o ON H 5526 5798 6055 6297 6524 6736 6934 7116 7283 7435 150 5514 5785 6042 6284 6511 6723 6920 7102 7269 7421 140 .5495 5767 6023 6265 64-91 6703 6900 7081 724a 7399 130 5471 5742 5998 6239 6466 6677 6873 7054 7221 7372 120 5440 5711 5967 6208 6434 6645 6840 7021 7187 7338 110 5403 5673 5929 6170 6395 6606 6801 6982 7148 7299 100 5360 5630 5885 6125 6351 6561 6756 6937 7102 7252 90 5310 5580 5835 6075 6300 6510 6705 6885 7050 7200 80 525''+ 5523 5778 6018 6242 6452 6647 6826 6991 7141 70 5192 5461 5715 5955 6179 6388 6583 6762 6926 7076 60 5123 5392 5646 5885 6109 6318 6512 6691 6855 7005 50 5048 5317 5571 5809 6033 6242 6435 66l4 6778 6927 40 4967 5235 5469 5727 5951 6159 6352 6531 6694 6843 30 4880 5148 5401 5639 5862 6070 6263 6441 6605 6753 20 4786 5054 5306 5544 5767 5975 6168 6345 65O8 6656 10 4686 4953 5206 544.3 5666 5873 6066 6243 6406 6553 0 4580 4847 5099 5336 5558 5765 5958 6135 6297 6444 Ill Inches of ¥ater Applied (11.46 inches of rain and soil aoisture) 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 v o o c 7570 7694 7802 7896 7975 8038 8087 8121 8139 190 7580 7704 7812 7905 7983 8047 8095 8128 8147 180 7584 7707 7815 7908 7986 8049 8097 8I30 8148 170 7581 7704 7812 7904 7982 8045 8O92 8125 8143 o 160 7573 7695 7802 7895 7972 8034 8082 8114 8132 V < 150 7558 7680 7787 7879 7956 9018 8O65 8097 8114 of-* fi. 140 7536 7658 7765 7856 7933 7995 80;% 8073 8090 S-t Inches of Water Applied (11.46 inches of rain and soil moistiore) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 200 8661 8918 9168 9410 9644 98?]. 10091 10302 10506 10703 190 8764 9023 9275 9520 9756 9985 10207 10420 10626 10825 180 8848 9110 9364 9610 9849 10080 10303 10519 10728 10928 170 8914 9177 9433 9682 9923 10156 IO38I 10599 10810 11012 o f-i 160 8960 9226 9484 10213 10440 10660 10873 11077 o 9734 9977 < o H 8988 Inches of water Applied (11.46 inches of rain and soil moistrire) 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 200 10892 11073 11247 11413 11571 11722 11865 12001 12129 190 11016 11199 11375 11543 11704 11857 12002 12140 12270 180 11121 11307 114B4 11655 11817 11972 12120 12259 12391 170 11207 11395 11575 11747 11912 12069 12218 12360 12494 16 0 11275 11464 11646 11821 11987 12146 12298 12442 12578 © ck c 150 11323 11514 11698 11875 12044 12205 12359 12505 12643 A 140 11352 11545 11732 11910 12081 12244 12400 12548 12689 ffi 11362 11558 117^ 11927 12100 12265 12423 •Hca 130 12573 12715 C 120 11353 11551 11741 11924 12099 12266 12426 12579 12723 (S 110 11325 11525 11717 11902 12079 12249 12411 12565 12712 co 100 11278 11480 11675 11861 12041 12212 12376 12533 12682 •I4-3 90 11212 11416 11613 11802 11983 12157 12323 12481 12632 44 O 80 11127 11532 11723 11906 12082 12250 12411 12564 W 11333 70 11023 11231 11432 11625 11811 11988 12159 12321 12476 1Pi 60. 10900 11110 11313 11508 11696 11876 12048 12213 12370 50 10758 10970 11175 11372 11562 11744 11919 12085 12244 40 10597 10811 11018 11218 11409 11593 11770 11939 12100 30 10417 IO633 10842 110/^ 11237 11424 11602 11773 11937 20 10217 10436 10647 IO85I 11047 11235 11416 11589 11754 10 9999 10220 10433 10639 10837 11027 11210 11385 11552 0 9762 9985 10200 10408 106 08 10800 10985 11162 11332 Table 26. Predicted yields of corn %rain in bushels per acre for selected nitrogen levels and irrigation ^ater applications Inches of Water Applied (11.46 inches of rain and soil moisture) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 200 98.4 103. 3 107.9 112.3 116.3 120.1 123.7 127.0 130.0 132.7 190 98.6 103.5 108.1 112.5 116.5 120.4 123.9 127.2 130.2 132.9 180 98.8 103.6 108.2 112.6 116.6 120.4 124.0 127.2 130.2 I33.O 170 98.8 103.6 108.2 112.6 116.6 120.4 124.0 127.2 I3O.2 132.9 c u o 160 98.7 103.5 108.1 112.5 116.5 120.3 123.8 127.1 130.1 132,8 u 150 98.5 103.3 107.9 112.2 116.3 120.1 123.6 126.8 129.8 132.5 R. %-t Inches of water Applied (11.46 inches of rain and soil sioisture) 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 200 135.2 137.4 139.3 141.0 142.4 143.5 144.4 145.0 145.3 190 135.4 137.6 139.5 141.2 142.6 143.7 144.6 145.2 145.5 180 135.4 137.6 139.6 141.2 142.6 143.7 144.6 145.2 145.5 170 135.4 137.6 139.5 141.2 142.5 143.7 144.5 145.1 145.4 160 135.2 137.4 139.3 141.0 142.4 143.5 144.3 144.9 145.2 150 135.0 137.1 139.1 140.7 142.1 143.2 144.0 144.6 144.9 1^0 134.6 136.8 138.7 140(3 141.7 142.8 143.6 144.2 144.5 130 134.1 136.3 138.2 139.8 141.1 142.2 143.1 143.6 143.9 120 133.5 135.6 137.5 139.2 140.5 141.6 142.4 143.0 143.3 lie 132.8 134.9 136.8 138.4 139.8 140.9 141.7 142.2 142.5 100 131.9 134.1 136.0 137.6 138.9 140.0 140.8 141.4 141.6 90 131.0 133.1 135.0 136.6 138.0 139.0 139.8 140.4 140.7 H 80 129.9 H 133.9 135.5 136.9 138.0 138.8 139.3 139.6 70 128.8 130.9 132.8 134.4 135.7 136.8 137.6 138.1 138.3 60 127.5 129.6 131.5 133.1 13 >.4 135.4 136.2 136.8 137.0 50 126.1 128.2 130.1 131.7 133.0 134.0 134.8 135.3 135.6 40 124.6 126.7 128.6 130.1 131.4 132.5 133.3 133.8 134.0 30 123.0 125.1 126.9 128.5 129.8 130.9 131.6 132.1 132.4 20 121.2 123.3 125.2 126.8 128.1 129.1 129.9 130.4 130.6 10 119.4 121.5 123.3 124.9 126.2 127.2 128.0 128.5 128.7 0 117.4 119.5 121.4 122.9 124.2 125.2 126.0 126.5 126.7 Table 27. Predicted yields of corn foraze in tons per acre for selected nitrogen levels and irrigation water applications Inches of Water Applied (11.46 inches of rain and soil moistizre) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 200 17.32 17.84 18.34 18.82 19.29 19.74 20.18 20.60 21.01 21.41 190 17.53 18.05 18.55 19.04 19.51 19.97 20.41 20.84 21.25 21.65 180 17.70 18.22 18.73 19.22 19.70 20.16 20.61 21.04 21.46 21.86 170 17.83 18.35 18.87 19.36 19.85 20.31 20.76 21.20 21.62 22.02 o 160 17.92 18.45 18.97 19.47 19.95 20.43 20.68 21.32 21.75 22.15 V 150 17.98 18.51 19.03 19.54 20.03 20.50 20.96 21.40 21.83 22.25 Pis 140 17.99 18.53 19.06 19.57 20.06 20.54 21.00 21.45 21.88 22.30 o (S3 130 17.97 18.51 19.04 19.56 20.05 20.54 21.01 21.46 21.90 22.32 3 -p o 120 17.91 18.46 18.99 19.51 20.01 20.50 20.97 21.43 21.87 22.30 I'r- g 110 17.81 18.37 18.90 19.42 19.93 20.42 20.90 21.36 21.80 22.24 h'" O -P 100 17.68 18.23 18.78 19.30 19.81 20.31 20.79 21.25 21.70 22.14 V; 90 • 17.50 18.06 18.61 19.14 19.65 20.15 20.64 2l.li 21.56 22.00 O W 80 17.29 17.86 18.41 18.94 19.46 19.96 20.45 20.93 21.38 21.83 C, § 70 17.04 17.61 18.17 18.70 19.23 19.73 20.23 20.71 21.17 21.61 60 16.76 17.33 17.89 18.43 18.96 19.47 19.96 20.45 20.91 21.36 50 16.43 17.01 17.57 13.12 18.65 19.16 19.66 20.15 20.62 21.08 40 16.07 16.65 17.21 17.76 18.30 18.82 19.33 19.81 20.29 20.75 30 15.66 16.25 16.82 17.37 17.91 18.44 18.95 19.44 19.92 20.38 20 15.22 15.81 16.39 16.95 17.49 18.02 18.53 19.03 19.51 19.98 10 14.75 15.34 15.92 16.48 17.03 17.56 18.08 18.58 19.07 19.54 0 14.23 14.83 15.41 15.98 16.53 17.07 17.59 18.10 18.59 19.06 117 Inches of Water Applied (11.46 inches of rain and soil violstvre) 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 200 21.78 22.15 22.49 22.83 23.14 23.44 23.73 24.00 24.26 190 22.03 22.40 22.75 23.09 23.41 23.71 24.00 24.28 24.54 ISO 22.24 22.61 22.97 23.31 13.63 23.94 24.24 24.52 24.78 170 22.41 22.79 23.15 23.49 23.82 24.14 24.44 24.72 24.99 a 22.55 22.93 23.29 23.64 23.97 24.29 24.60 24.a3 k < 150 22.65 23.03 23.40 23.75 24.09 24.41 24.72 25.01 25.29 u 22.70 23.09 23.,% 23.82 24.16 24.49 24.80 25.IO Î-1 a :L30 22.72 23.12 23.49 23.85 24.20 24.53 24.85 25.15 25.43 S 120 22.71 23.10 23.48 23.85 24.20 24.53 24.85 25.16 25.45 o ^ 110 22.65 23.05 23.43 23.80 24.16 24.50 24.82 25.13 25,42 C o 100 22.56 22.96 23.35 23.72 24.08 24./I2 24.75 25.07 25.36 -p 90 0 22.05 22.46 22.86 23.25 23.62 23.98 24.32 24.64 24.95 P-, 60 21.80 22.22 22.63 23.02 23.39 23.75 24.10 24.43 24.7b 50 21.52 21.94 22.35 22.74 23.12 23.49 23.84 24.17 24.4? 40 21.19 21.62 22.04 22.44 22.82 23.I9 23.54 23.88 24.20 30 20.83 21.27 21.68 22.09 22.47 22.85 23.20 23.55 23.87 20 20.43 20.87 21.29 21.70 22.09 22.47 22.83 23.18 23.51 10 20.00 20.44 20.87 21.28 21.67 22.05 22.42 22.77 23.IO 0 19.52 19.97 20.40 20.82 21.22 21.60 21.97 22.32 22.66 118 Summary Because data werenot available for estimating the parameters of the true response functions of crops to irrigation water, the parameters of a quadratic equation were estimated for corn using irrigation water and fertilizer from the results of an experiment designed for the pur pose of production surface estimation. The results of the experiment at Fort Collins, Colorado, indicates the response of corn to follow traditional economic thinking. The quadratic function was fitted to the data and the statistical analysis indicated significant response to the variables. About 65 percent of the variation in grain yield was explained by the regressions. The for corn forage production vari ation was .60. 119 ALTERNATIVE METHODS OF VALUING IRRIGATION WATER The primary objective of this section is to outline the theoretical basis for resource demand functions. Although the price of irrigation water can only be determined at the equilibrium of supply and demand, the valuation takes place by the use of the theory of marginal pro ductivity. The theory of marginal productivity at most analyzes the factors affecting the demand for an input of production. The price of the input depends also upon the conditions of supply. Therefore, one can regard the theory of marginal productivity as a theory solely of the demand for factors of production. Economic evaluation of the potential use and development of water resources requires estimates of the marginal value product of water. For example, marginal values are needed to assess the benefits from potential projects to supply irrigation water. They also are needed to determine the optimal allocation of existing water supplies among competing uses. The demand curve for a factor of production by a particular group of demanders shows the maximum quantity of the factor that will be pur chased per unit of time at each price of the factor. Presently, there are three direct approaches for estimating water values in irrigation: budgeting or residual imputation, linear pro gramming or activity analysis, and classical production function analysis. The first method, budgeting, is an historically widely used method 120 of factor valuation in which the product value imputed to a given re source is taken as the difference between the total value product and the sum of algebraic products of the quantities of resources multiplied by their market prices. This method draws heavily on the usually unre alistic assumptions that there are constant returns to scale and that the marginal value product of all factors is equal to the market price except for the one being estimated. In addition, the estimate provides the total or the average return to the factor considered rather than the marginal. The linear programming approach, the second method, approximates the marginal productivity by the average productivity. This method, however, may be applied only when one can assume that the average pro ductivity, and consequently the marginal productivity, are linear func tions. The third method used extensively for estimating marginal produc tivities is based on the derivation of empirical production functions. The theoretical production function model is a basic tool in microecon- omic theory. Considering a firm as an effective decision making unit, one can confront the firm by one or more production functions determined by technical conditions. For a single product, Y, which is dependent upon several factors, the function is: Y = fCx^, Xg) •••' When some of the factors are fixed, the function is: Y - f(x^, X2, • • •, Xj j Xj , Xj^2) •••? ) 121 where x , x , x are the non-variable inputs- j+1 j+2 n The production function being given, one may derive basic quanti ties for marginal analysis, i.e., isoquants, isoclines, marginal rates of substitution, and the marginal physical productivity of factors. Attempts to estimate empirically the marginal productivity of resources may be traced back as far as Ricardo. However, it can hardly be claimed that a technique beyond criticism has been achieved yet. Budgeting or residual imputation* Budgeting has been used by research and extension economists for many years. Budgeting is also used by agencies of the federal govern ment to value irrigation water; however, the residual return after other inputs have received their payment is usually a composite payment to water, plus to several other nonmarket inputs such as management and family labor. It is extremely difficult to separate the marginal value product for a single resource by using the budgeting approach. About 1920, attempts at a forward looking process in farm manage ment brought about the forerunner of the present budget process. Com plete and partial budgeting procedures evolved naturally as an outgrowth of the theory. But budgeting existed on the fringe of theoretical re *Much of the following discussion is summarized from the following research bulletin: Irving F. Fellows, editor. Budgeting—tool of re search and extension in agricultural economics. Connecticut (Storrs) Agricultural Experiment Station Bulletin 357. 1960. 122 spectability. It was evident that the marginal concept could be applied, but problems relative to other major assumptions were answered less satisfactorily. The critical question in point is this: Can budgeting simultaneous ly include all the conditions leading to firm equilibrium? Alternatives of all kinds can be compared by successive budgets. This points to one of the weaknesses of the budget process. As a trial and error method, it is very time consuming. However, the knowledge and skill of the manager facilitates his choosing those alternatives which have the greatest likelihood of success. Estimates of future expectations are needed to construct budgets. Some skill is needed to evaluate risk and to incorporate this concept through discounting, although this problem can be minimized by a standardized form and procedure. A major disadvantage of the budgeting procedure is its time consuming character when alternative situations are tested, as they must be under the nearly universal condition of limited resources. The objective of budgeting is to discover an optimuir. allocation of resources- Conceptually, one is seeking a point in n-dimensional space. Although one can seldom hope to discover the exact optimum point, there is great danger that the process may not move close to the optimum. Fre quently, the process is used merely to discover a new resource combina tion which is superior to a previous combination. 123 Several reasons exist for the frequent failure of the process to approach the optimum point. One reason is that many alternatives can not be thoroughly tested because adequate technical information is not available or is not applicable to a specific situation. This lack of information affects all parts of the production function. An economic analysis of resource use depends upon a knowledge of the change in output resulting from a change in the level of the input. Another reason for failure is that the interaction and side effects at various levels of inputs are often confounded and unidentifiable. Recognition of the existence and contribution of other resources may be inadequate. A major step in overcoming the inadequacies of technical data lies with the design of experiments. Discovery of an approximate optimum point often depends upon the recognition and exploitation of some resources which are relatively fixed for the firm. To give guidance and to stimulate the budgeter to fully exploit fixed resources, standards of performance for labor, equipment and land resources can be used. Basically, the budgeter needs complete input-output information. Budgets are forward looking and are based upon estimates of the future changes in costs and returns. Estimates are needed for the future prices of production resources and of farm products. Expectations are needed for the technical processes of production. Such difficult prob lems as the possible development of new and superior techniques must be considered. Predictions cannot be made with certainty, for too many 124 variables occur in areas in which the budgeter cannot have complete knowledge. Regardless of the problems, estimates must be made, and they should be as realistic as possible. One should realize, however, that they will not be entirely accurate; for planning by any method is im possible without estimates of future events. The budgeting technique is a useful tool in seeking an optimum re source combination with a particular situation. To be used effectively, certain problems must be overcome. Recognition of these problems and uniform solutions of them will increase the effectiveness of this tool in farm management research. An agency charged with allocating water generally approaches the problem by budgeting several different farm situations, usually with and without irrigation. From these budgets a residual return to water is derived. The return is used to derive the net benefit of irrigation and the repayment schedule for the irrigation structures. Such an ap proach, while being useful, does not make the best use of the informa tion which is available. Production function analysis would allow the incorporation of basic economic principles into the decision making framework. It would permit estimation of the limits of resource use and the substitution of resources for one another. As most production functions do, it would provide a non-linear response which corresponds to agronomic and economic theory. In contrast, budgeting provides no mechanism for resource substitution and implies that one can increase output indefinitely simply by adding to the resource inputs. 125 A farm production Cuaction Typically, a farmer operates a fixed area of land with which, he combines other inputs to obtain production. An increase in the supply of water to a farm with fixed land area will generally result, at some stage, in decreasing marginal productivity of water in the aggregate function. Given information on the water production function for ir rigated crops, a rational farmer will allocate water to the various pro- r " ductive activities according to the equimarginal principles. As a re sult, an aggregate production function of the entire farm is obtained, with total output value as a function of the total input of water and other resources. However, information available so far regarding the empirical water production function of various crops is rather scanty. Even, if the water output relationships for individual crops are restricted to constant water input per land area (because of lack of information on the effect of variation in irrigation intensity on yield response), it can be shown that an aggregate water-output production function for the farm as a whole does exist. To illustrate, consider a farm firm which has only two limiting resources, land and irrigation water. The water requirements for the three crops are fixed per acre of land. In Figure 22 the constant returns to scale from each crop is shown by the rays from the origin, Cj^, C2 and . The isoquants, the broken lines and I^, represent succes sively higher net output in value terms. L 0 Water rigure 22. Combining land and water with three crops to maximize income 127 the combinations of water and land necessary to produce a given level of net output using various crops. From the way Figure 22 has been drawn, it is obvious that is a land intensive crop, is water intensive, and C2 is intermediate between. The land-water ratio differs between any two crops but is always the same for any particular crop. Typically, a farmer controls a given piece of land to which he can apply varying amounts of water. If his land is fixed at OL, as water varies from left to right along the line LL', the land-water ratio de clines as a greater amount of water is applied to a fixed land area. Decreasing returns to water are experienced as evidenced by the increas ing distance between the isoquants as one progresses to the right. Ob viously, the farmer will try to reach the highest isoquant which is permitted by his fixed resources. If the farmer has OW^ water available, Ig net output will be realized; with OW2, net output will increase to I3; and with OW3 amount of water, I^ returns will be gained. The analysis above can be expanded to include any number of pro duction agents and production functions estimated through the use of linear programming techniques. Given feasibility of the production function approach, problems of water allocation can be solved using conventional economic theory, and the demand for water by a farm can be derived directly from its aggregate production function. Until pro duction coefficients become available which will allow application of conventional production function analysis, economists must make use of 128 the next best method. Problems such as the farm production function illustrated above are well suited to analysis by the use of linear programming techniques. Attention will next be given to the uses of linear programming in farm resource allocation problems. Linear programming Both budgeting and linear programming involve the same basic as sumptions; in fact, linear programming can be called systemized budget ing. However, linear programming does have the advantage of isolating a single optimum without resorting to the time consuming trial and error budgeting approach. It allows the marginal value product for each limiting factor of production to be determined simultaneously with the system of equations being solved. Since computer facilities are now readily available, linear programming provides answers much more economically and generally with a higher degree of precision than budgeting - A linear programming problem has three quantitative components. The first is an objective function; for irrigated agriculture, the objective function might be to maximize income or to minimize costs. The second component of a linear programming problem is an alternative method or process for attaining the objective- If alternative methods of combining resources are not available, there would be no problem to solve. The third and last component is the resource restraint or other restrictions. If restrictive quantities of resources are not available, a linear programming problem would not exist. 129 The term linear refers to the fact that straight line relation ships are employed in linear programming and that only equations with variables in the first power represent linear relationships. The input-output coefficients used are assumed constant, as are the prices paid for resources or received for products. The term linear program ming arose from application of a particular mathematical procedure, based on linear relationships and inequalities, to choice problems. Inequalities arise when a plan is determined which does not require using the supply of all available resources and guarantees that the amount of any activity or commodity produced will be equal to or greater than zero. Therefore, the quantity of the resource used will be less than or equal to the quantity available. Linear programming techniques involve the maximization or minimiza tion of a linear function subject to the linear inequalities. The linear function ordinarily is a profit or a cost function. For example, the function P = P Y + P Z o y z is a profit function where is profit, P^ is the net price per unit of Y, and is the net price per unit of Z. This is a linear function and if the function is maximized, profit will be maximized. In the general form of programming, one can: n maximize Z = S C- X- i=l subject to 130 a^l 4. a^2 Xg ^ \ 321 * ^22 ^2 * '" •*" ^2n \ ~ ^2 ^ml Xl + *m2 *2 + "" + ^mn and each X^ - 0 where n activities are considered and m resources are restricted in use. In matrix form would be the following: max CX sub to AX - b and X - 0 . Comparing linear programming with budgeting Linear programming is a mathematical method of budgeting and can be used as a substitute for the older budgeting method because it allows computations at lower cost and with a savings of time. Budgeting and linear programming are not distinctly different procedures but are the same procedure with differences allowed in the number of opportunities considered and the calculations involved. Both methods use the concepts of linearity, and budgeting should be considered the forerunner of linear programming. In complete budgeting, the investigator typically compares two organizations of a farm. The implicit assumption is that the production possibility curves are linear and the substitution rates for products 131 and resources are constant. Unlike budgeting, linear programming al lows consideration of many organizations and alternatives. There is, of course, no theoretical reason why budgeting could not also consider the same alternatives, for if enough alternative budgets are considered, the assumptions of linear programming are duplicated. However, the con sideration of so many budgets becomes very burdensome and hence is com putationally infeasible. Under linear programming models, the use of computers allows the increase in alternatives considered. Some analysts believe budgeting to be a more practical tool since they can incorporate subjective knowledge of the situation and make better use of field experience. However, the same can be done for linear programming. Many simple decision problems can be answered as easily by budget ing as by linear programming. However, when the number of restricting resources and possible activities and techniques is large, linear pro gramming is the efficient procedure for imputation processes. Linear programming allows the marginal value product for each limiting factor of production (i.e., the value of water) to be determined simultaneously as the system of equations is solved. Simplex tableau for a linear programming problem The objective of this section is to indicate how price data and technical coefficients may be used in setting up a simplex tableau. The simplex tableau used is an example of a generalized maximization problem. 132 The ultimate prupose of the farm budgets is to determine the op timal organization of farms for maximization of profit and to impute the value of a limited resource. The linear programming example uses two alternative livestock enterprises and thus allows the possibility of none, one or two livestock enterprises in the optimal solution. Both sales and purchases of alfalfa and barley are allowed. The linear pro gramming example deals with only one class of land (L^) but includes transfer activities that permit land leveling and irrigation; hence, it allows the possibility of additional types of land, namely leveled land (L2), irrigated land and irrigated and leveled land The objective of the linear program is to maximize the value of barley, alfalfa, beef and dairy produced on the farm, subject to the re source restrictions on land, capital and labor. The basic data for the linear programming example are given in Table 28. The input-output coefficients and resource requirements per unit . of crop produced are given in the simplex tableau, Table 29. The dis posal activities serve two purposes. First, linear inequalities, which may exist if the plan does not use all the supply of an available fixed factor, may be changed to equalities by using disposal activities. The levels of the disposal activities are called slack variables, and allow non-use of resources. The second use of the disposal activities is the basis for the initial solution. The real activities are those which are bought or sold in the market. Rows P22 and ^23' barley and alfalfa respec tively, represent intermediate products which may be fed to beef cattle or dairy cattle. Columns and alfalfa buying and barley buying 133 Table 28. Data for programming Item Barley (Bly.) Alfalfa (Alf.) Yield per acre on (bushels) (tons) ^IB ^lA Yield per acre on (bushels) (tons ) ^2B ^2A Yield per acre on (bushels) (tons) ^3B ^3A Yield per acre on (bushels) (tons) per unit ($) (bushels) (tons) Price ^A Gross yield per acre on ($) ^A^IA Gross yield per acre on L2 V2B %A Gross yield per acre on ^B^3B ^A^3A Gross yield per acre on %^4B %A Variable cost per acre on ($) SB ClA Variable cost per acre on L2 SB SA Variable cost per acre on ^3B C3A Variable cost per acre on C4B SA Net return, above fixed costs, per acre ($) ^1^ VlB'^lB *5= :A?lA-CïA Net return, above fixed costs, per acre ($) Net return, above fixed costs, per acre ($) Net return, above fixed costs, per acre ($) ^4" ^B^4B"*^4B ^8 ^A^A'^^a 134 respectively, represent a source of barley and alfalfa which may be fed to cattle. Column Pg represents a land leveling activity which will convert L^, land not leveled or irrigated, into , leveled land- Columns P^g and P^^ serve similar purposes. Some of the input-output coefficients are positive while others are negative. Positive coeffi cients result when units of a resource (row) are used by an activity (column). For example, the intersection of row P^g and column P^ has a 1, which means that 1 acre of land which is not leveled or irrigated is required to produce one unit of barley (Y^g in Table 29). The pro duction of one unit of barley thus becomes available to the resource row P22 and is represented as -^22 1 because it adds to the resource, which may later be used by the beef cattle or dairy cattle. (Hence, A and A„„ have a positive coefficient.) 22 g 12 Zz J1o As mentioned above, the row designated C is a row of net profits- through Rg were previously derived. One problem to avoid is double counting. Since the net revenue of barley and alfalfa has been imputed, the analyst must calculate the net revenue to activities P^2 ^rid P^^, beef and dairy, and not use market price.^ Because the cost of barley and alfalfa as inputs have already been subtracted from beef and dairy net revenues, R^2 respectively, then and R^^ are zeros. In the case of Pg, P^g, and P^^, the land leveling, irrigating and combined land leveling and irrigating should have negative net revenues, Rg, R^g, and Rjj' since increases in these activities decrease the profit function, would, in the initial coefficient matrix, have the same value as A2Q g, 0. Heady and Wilfred Candler. Linear programming methods. Ames, Iowa, Iowa State University Press. 1958. p. 112. Table 29. Simplex tableau Real activities C Ri R2 *3 R4 % ^6 ^7 ^8 R9 Resource Resource or or activ Bly. Bly. Bly. Bly. Alf. Alf. Alf. Alf. Li • Lj • activity ity level Pi P2 P3 Pe P? Pg P9 Cs B 0 (Lj^: Land not leveled or irrigated) 100 ^16 1 1 1 0 (Lg: Leveled but not irrigated land) Pl7 0 1 1 -1 0 (L : Irrigated but ^ not leveled) 0 1 1 ^8 0 (L.; Irrigated leveled land) 0 1 1 0 Capital 130 ^2 0 *20,1 *20,2 *20,3 *20,4 *20,5 *20,6 *20,7 *20,8 *20,9 0 Labor 400 ^21 *21,1 *21,2 *21,3 *21,4 *21,5 *21,6 *21,7 *21,8 0 Barley 0 P22 •*22,1 -*22,2 "*22,3 "*22,4 0 0 Alfalfa ^2 3 •*2 3,5"'*23,6' '*23,7' *23,8 Z 0 0 0 0 0 0 0 0 0 0 Z-C -Rg -Rg -Ri -R2 -*3 -R4 -^5 -R? -R9 Table 29. (Continued) Disposal activities "lO «11 «12 «13 «14 «15 0 0 0 0 0 0 0 0 L.L. â nd AIE# Bly« 1j2 ^ C LiSb# Bly» AIE# Ir. Ir. Beef Dairy buy buy ^10 ^11 ^12 ^13 ^14 ^15 ^16 Pl7 Pl8 ^19 ^20 ^21 ^22 ^23 ^ 1 •*20,10*20,11 *20,12 *20,13 *20,14 *20,15 I30/A20 4 1 *21,12 *21,13 400/*2i , 1 - - *22,12 *22,13 •*22,15 1 -- *23,12 *23,13"*23,14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 "*10 -*11 -*12 -*13 -*14 -*15 137 which is the capital required to produce one acre of land leveling activity. The assumed initial level of resources was 100 acres of land which was not leveled or irrigated, 130 units of capital with each unit equaling one hundred dollars, and 400 days of labor. Economic interpretation of simplex procedures If the model is correctly constructed and the computation steps are followed, then each iteration brings the analyst closer to a solution. In the simplex table, the quantities in the B column are disposal products. However, in subsequent iterations the values are either marketable products or disposal products. The values in the input-output section of the table for any given row and column are transformation rates. These rates specify the amount of the activity represented by the rcw which must be sacrificed to gain one unit of the activity represented by the column. In the simplex tableau one might assume that is the largest net revenue. The calculated row Z is a row of zeros. The values of the Z row for each column Pj, respectively, represent the value of other enter prises sacrificed as Pj is increased by one unit. Thus, the Zj is the opportunity cost for the jth activity. The marginal value product is Z.-R. and is shown in the Z-C row. Since R- is the net revenue from the 3 3 J Pjth activity, if the opportunity cost is less than the net revenue from increasing the activity, then the profit function may be increased by selecting that activity to put into the basis (b column). Since has 138 the largest negative Zj-Rj value in the tableau, it is selected to enter the basis. The R column is calculated as prescribed, and one sees that column should replace row P^g. Thus, production of barley on irri gated leveled land is the first activity to enter the basis, even if it is at zero level of production. If the computations were carried further, the P^^^ column would be the next one to enter the basis and would replace row Pjg- This would cause row P^ to remain in the basis but be at the level of 100 acres. Thus, it could be shown how the transfer activity operates. Positive Zj-Cj values indicate that an increase in the jth real activity will decrease profits. However, in the case of the disposal activities, positive Zj-Cj values (shadow prices) represent the marginal value product of the corresponding resources. A one unit increase in the resource will increase profit by the amount of Zj-Cj. Only re sources which are scarce have positive marginal value productivities. Thus, by computing the solution one can determine the optimal out put combination and know the possible gains from acquisition of scarce resources. Other applications of linear programming The concept of shadow prices, the marginal value product of a fixed resource, has been dis cussed. Using parametric linear programming, one can change a resource constraint, such as leveled land or water level, and determine the in fluence of this change upon farm income. Thus, linear programming can be used to evaluate the influence of institutional and physical constraints 139 of land or other resources upon farm income. Directly, an activity can be devised to provide the portion of income attributable to irri gation development. When prime interest is in this quantity, it can be computed directly as the dual solution of the programming model. Estimating the demand for irrigation water Estimating the demand curve for irrigation water is a matter of great importance in most arid regions and is receiving increased attention in many humid areas. The attempt here is to illustrate how to estimate the static normative demand for irrigation water for individual farms. If there are changes in any of the three major components of a linear program, the optimal solution may change. For a resource for which a positive shadow price exists (the marginal value product is positive), the researcher may want to increase the quantity of the re source made available in the program. Since water is an important constraint on irrigated farms, it may be useful to find out how farm income changes as additional water is made available to the farm enter prise. The static-normative nature of a demand function estimated may not be very indicative of what farmers will do in the short run if water prices change drastically. However, farmers will tend to adjust to what the analysis indicates they should do. The value of the objective function may be greatly influenced by variable price programming as the price of a resource is varied. The derived normative demand function for water describes the quantity of 140 water the farmer should purchase to maximize income at each set of prices- A summary of solutions obtained for a beef farm in central Utah is given in Table 30. The price of water was varied from $10 to $0 per acre foot. The value of the objective function is given in column 3. As more units of the water resource are purchased, the objective function increases be cause: (1) more water is used, giving greater yields; and (2) the price of water is lowered. The influence of lowering the price of water upon the objective function is given in the following example. When water was $10 per acre foot, 198.19 units of water were purchased and the value of the objective function was $3842.51. The objective function increased by $205.12 to $4047.63 when the price of water was lowered to $9.00 per acre foot. The increase in the quantity of water used was 19 acre feet. The initial 198.19 acre feet of water is now purchased at a price of $9.00 per acre foot rather than $10.00 per acre foot. This $198.19 accounts for a large part of the $205-12 gain in the objective function. Only $6.93 is at tributable to more water being applied to crops. Actually, the additional 19.0 acre feet of water increased the objective function by only $6.93 or $0,365 per acre foot. The normative demand curve for water is shown in Figure 23 as a step function. The interpretation of the normative demand curve is that, for any given price of water, the demand curve shows the optimum quantity of water a farmer should buy. Alternatively, for any given quantity of water, the demand curve shows the highest price the farmer can afford to pay for water. 10.00. 9.00. 8.00 7.00 8 ^ 6.00. a> t- u n 5.00 S 4.00. 3.00, 2.00, 1.00. ONfi-r III I II" Vf csl o r-j o vO VÛ o o o -4* O O O LTN 00 O m CO m o J- lA o mm O unco Figure 23. Normative demand curve for water delivered at farm Table 30. Summary of solutions to variable price programming for water delivered to the farm Value of Change in Increase of Value Price of Quantity objective water Increase in Savings due function due per water of water function purchased revenue to price change to more water acre foot ($) (acre foot) ($) (acre foot) ($) ($) ($) ($) 10.00 198.19 3842.51 19.00 205.12 198.19 6.93 .365 9.00 217.19 4047.63 0.00 217.20 217.19 0.00 m# ^ — 8.00 217.19 4264.83 11.40 221.22 217.19 4.03 .353 7.00 228.59 4486.05 5.05 230.23 228.59 1.64 .324 6.00 233.64 4716.28 89.77 306.35 233.64 72.71 .810 5.00 32 3.41 5022.63 0.00 323.41 323.41 0.00 » o O 323.41 5346.04 0.00 323.41 323.41 0.00 3.00 323.41 5669.45 49.86 345.31 323.41 21.90 .439 2 .00 373.27 6014.76 2.24 373.84 373.27 .57 .254 1.00 375.51 6388.60 2 .95 37 7.88 375.51 2.37 .803 0.00 378.46 6766.48 Classical production function and marginal analysis The analysis of the demand for factors, or in this case water, is closely related to the analysis of supply of products and is really only another way of looking at the same thing. The statement that a firm seeks to equate marginal factor cost to marginal value product is another way of saying that it seeks to equate marginal cost to marginal revenue. When the marginal productivity theory first achieved popularity just before the turn of the century, some economists attempted to use it as a basis for moralizing. It was even suggested that since the analysis showed that every input was paid the value of its marginal product, the distribution of income under free competitive capitalism must be morally right and just.^ As William J. Baumol explained in his book: "The marginal productivity theory is an empirical hypothesis, and from it no legitimate evaluative conclusion can be drawn without the careful and explicit addition of a set of value judgements which can serve as the 2 criteria of good and evil." Classical marginal analysis is couched in terms of production func tion analysis and can be conveniently described in mathematical terminol ogy. The criteria of efficient resource allocation are the same for all production functions irrespective of the algebraic form. Supposing the farm firm is applying irrigation water to a fixed unit of land producing one particular crop with all other factors non-scarce. ^J. B. Clark. The distribution of wealth. New York, N.Y. The Macmillan Company. 1899. Chapter 1. William J- Baumol. Economic theory and operation analysis. 2nd ed. Englewood Cliffs, N.J., Prentice-Hall, Inc. 1965. p. 404. 14U What then is the condition for equilibrium? The problem would be one of profit maximization: T = TR - TC T = Py[f(Xi)] - Px^X, where Y = fCX^) ^' Py(f'Xl) - % - ° PyCfXi) = Pxi Py • Mppxi = PXI mPPXl M?PXi= Pxi MR= MC . For profit maximization the firm would equate marginal revenue with marginal cost or alternatively and equivalently equate marginal value product of water to the price of water. By multiplying the price of the output, Y, by the marginal physical product of water curve, the marginal value product curve is obtained. This curve is the firm demand for water schedule. The schedule is the locus of various quantities of water the firm would be willing to purchase at varying prices. Mainly, the marginal productivity theory is a way of organizing the considerations that are relevant to the demand for a factor of production. In analyzing the supply curve of a product, interest is in tracing the effect of changes in the demand for the product under given conditions on the factor markets. In consequence, attention is directed to the out 145 put of the firm or industry, and taken for granted are the changes in the quantity of the various factors of production employed and in their prices, as demand for the product and, with it, output of the product change. In distribution theory, interest centers in the factor markets, and so concentration is on a different facet of the same adjustment by the firm. Consider the following equation: py . MPPX^ = PX^ . It shows a relation between the price of and its quantity. For each price of it shows the quantity of X^ that would have a marginal product whose value would be equal to that price of X-^. It is tempting to interpret this as the demand curve of the firm for Xj^. But this is strictly correct only in one special case, that in which the firm is not free to vary the quantity of any factor other than X^, i.e., all other factors are "fixed". In that case, the only adjustment the firm can make to a change in the price of X^ is to change the quantity of X^ employed. The firm will move along the marginal product curve for X^ until the value of the marginal product is equal to the new price of Xj, and this curve will be its demand curve. If the constraint is relaxed upon the "fixity" of. other factors and allows, for example, Xg to vary and assumes Xg is purchased competitively, then a different demand curve for X^ is traced out. Suppose the price of X^ falls so that the firm desires to purchase more of X^ until the equi librium condition is satisfied. The conditions for the other factors are 146 now not satisfied. The reason is that the marginal product of the other factors depends upon the amount of used in production. Some factors will be close substitutes for X^, and their marginal products will de crease by greater use of X^. Other factors will have their marginal product increased by the increased use of Generally, one would ex pect the second effect to be dominant. In any case, the firm will want to change the amount of other factors used. Such adjustments will in turn change the productivity of X^. A final equilibrium will be achieved after all adjustments have taken place and all equations are satisfied. At this final position, the price of X^ is equal to the value of its marginal product, but this point will not be on the original curve. Ac tually, the marginal product curve is for given fixed quantities of other factors, while the actual demand curve is for fixed prices of variable factors. Euler's theorem There is a standard mathematical result, called Euler's theorem, which explains that if a production function involves constant returns to scale, the sum of the marginal products will actually add up to the total product. The proof depends on the total differentiation proposition which states that if one has the function Y = f(L, W ) and that if, in turn, L and W are both functions of some variable t, i.e., L = F(t) and W = G(t), then: d Y jf dL èf dw d t " • d t + aw ' d~r • 147 The result is intuitively plausible and states that the effect of a change in t on Y is composed of two parts: the part which is transmitted via the effect of t on L; and the part which is transmitted through W. Thus, the latter is represented by the expression (<) f/j .W )(d W /d t). Then (d W /d t) is the change in W produced by the increment in t, and ( change in W.^ To derive Euler's theorem, one notes that for a linear homogeneous production function Y = f(L, W ) he has for any K: KY = F (KL, KW) Taking the total derivative of KY with respect to K (i.e., setting KY = Z, KL = X, KW = y and K = t) in the formula for the following is obtained: d KY _ à f d KL Ù f d KW d K KL * d K ' d K or j ^ Y - ÙKL L JKW W " Since this result holds for any value of K, it must also be valid for K = 1 so that: Y = T L - W L <5 W • This is Euler's theorem for the linear homogeneous production function Y = f(L, W).^ The proof holds for any number of inputs. Since èf/âL ^Ibid., pp. 40U-.U05. 148 is the marginal product of land and ôf/^ W is the marginal product of water, the equation states that the marginal product of land multiplied by the units of land plus the corresponding total payment to water exactly equals the total product, Y. The basic postulates of Euler's theorem are that: (1) each input is paid the value of its marginal product; and (2) total output is just ex hausted. Since these conditions were met by production functions homogene ous of degree one, it was generally assumed by reverse reasoning that all production functions were of this type. Economists have found that not all production functions are homogeneous of degree one, and in addi tion have discovered that the assumption of homogeneity is not necessary for the fulfillment of the second postulate of the marginal productivity theory (the condition of product exhaustion). The condition of product exhaustion is equivalent to the condition that maximum long run profit equals zero. Multiplying the previous result through by the price of Y gives: PY = L + p ^W. J i. ^ W \ ^ S f Substituting = ? "3^ and Py = from the condition for profit maximization results in: PY = P^L + Py W. Long run total cost equals long run total revenue, which leads to the conclusion that long-run profit equals zero regardless of the price. However, the postulate that each input is paid the value of its marginal product need not be fulfilled under the above conditions. There appears 149 to be nothing sacred or moral about the equation of input price to the value of its marginal product, but under certain conditions it is the way for the farm firm to maximize profits. These conditions are that both the factor and product markets must be perfectly competitive. The problem of resource valuation is basically and fundamentally one of allocating or imputing the total product forthcoming in a single pro duction process to each of the several resources involved. The product or reward to one factor of production cannot be established accurately except as the rewards for other factors are accurately reflected. When the market rate is less than the actual productivity of resources, part of the return to other factors is imputed to the one receiving the residual. If the market rate is greater than the productivity, the reverse holds true. Actually, there is no basis for allocating the residual surplus or deficit to any single resource. The Euler system suggests that if the value or marginal productivity of one factor is to be estimated apart from the market, the reward of other factors might equally be estimated apart from the market. Imputing the value of irrigation water using empirical production functions For ease of computations for economic analysis a transformation was made of the coefficients of the estimated production functions for corn in Colorado based on inches of water applied. The coded X^'s were re- — X2 — placed by X^-X^ and the coded X^'s replaced by ^ - X2 150 where is the mean of the actual values after the actual values were divided by 50. The new functions are as follows. %grain=4579.45 + 549.24%! + lO.QSXg - 29.96X^ - .0314X2^ + .OôZôX^Xg and Tforage=7114.91 + 605.SIX^ + 26.76X2 - 15.22X^2 _ .OSSXg^-.416X^X2 where grain and forage are measured in pounds per acre, X], is inches of water and Xg is pounds of nitrogen. The expected yields with any given inputs of water and nitrogen may be calculated directly with the trans formed functions. Using information from the production function, a demand function for either of the inputs can be derived. Given a change in the price of an in put- adjustment in the quantity of the input employed would be specified by the profit maximizing equation a Y = Px ^ ^ Py where ^ Y/c>X is the partial derivative of the production function, is the price of the input, and P^ is the price of the output. The partial derivative of the quadratic function for Xj is »! - *1 * bl2X2 ' Equating to the price ratio and solving for X^ gives "l • "12*2 - Fx, Py-1 4 = nr. 11 151 and similarly for X2 we get .1 h + bi2*l The demand for the input is specified as a function of the price of the input, the price of the output, the constants of the production func tion, and the amount of the other input used. If the interaction term of the original function was zero the inputs would be independent of one another and the use of one would not depend upon the use of the other. The interaction terms of the equations estimated were not significant, thus the demand function could be simplified. However, for illustrative purposes the interaction term has to be carried along as though significant. In order to utilize the demand function, the quantity of the other input used must be specified as well as the prices of the input being considered and the output. As an example the price of corn grain will be 1.78 cents per pound, the price of irrigation water will be 1 dollar per acre inch and the level of nitrogen use will be 100 pounds per acre- Inserting the constants of the production function we have 549.24 + .0626(100) - (1.00/.0178) X 1 2 (29.96) when solved gives 8.33 inches of water. For forage production with the same water price, nitrogen at 7 0 pounds and a forage price of 1.0 cents per pound or 5 dollars per ton, the profit maximizing use of water would be _ 605.53 - .416(70) - (1.00/.01) X. 15.6 1 2 (15.22) inches of water. 152 Similarly for the nitrogen input in grain production, if the water use is assumed to be 7.0 inches with a grain price of 1.78 cents per pound and a nitrogen price of 10 cents per pound, the profit maximizing use of nitrogen can be found which with varying prices of nitrogen will trace out a static demand function. 10.93 + .0626(7.0) - (.10/.0178) _ *2 ° 2 (.0314) ° " pounds of nitrogen. For forage production with the same water use and nitrogen price with a forage price of 1 cent per pound, the solution would be 26.76 - .416(7.0) _ (.10/.01) ^2 ~ 2(.095) =73 pounds of nitrogen. Table 31 shows the quantities of water which would be demanded for various prices of water and varying applications of fertilizer when pro ducing grain. The demand curves from the data in Table 31 are given in Figure 24. A more generalized, less static, demand function would be de scribed by the simultaneous solution of the previous individual demand equations. The generalized simultaneous solution was given in Chapter two. The equations should be used in pairs with each input being a function of the input prices, output price and the constants of the produc tion function. The use of an input is dependent upon the level of use of the other input only indirectly, because as any price varies, the profit maximizing quantities of both inputs will change. Once again; assuming the previous prices of 1.78 cents per pound of corn grain, 1.0 cents per 153 Table 31. Quantities of water to use for profit maximization for various water prices and nitrogen applications with a grain price of 1.78 cents per pound Water price Pounds of nitrogen Acre inches per acre inch per acre of water .00 0 9.166 50 9.218 100 9.271 150 9.323 200 9.375 .50 0 8.699 50 8.751 100 8.803 150 8.856 2 00 8.908 1.00 0 8.233 50 8.284 100 8.336 150 8.388 2 00 8.441 1.50 0 7.764 50 7.817 100 7.869 150 7.921 200 7.973 2.00 0 7.297 50 7-349 100 7.402 150 7.454 200 7.506 pound for corn forage, $1.00 per acre inch of water, and 10 cents per pound for nitrogen; the equations can be solved for the level of input use. For corn grain we have = 8.32 inches of water X2 = 92.71 pounds of nitrogen and for corn forage 2,00 Simultaneous solution demand _c u c 0) L_ o m 9.0 ''•ere inches of water Figure 2k. Static water demand curves with constant nitrogen and grain prices with three levels of nitrogen application (The flatter curve represents water demand allowing nitrogen application to vary simultaneously,) 155 = 14.4 inches of water X2 = 53.5 pounds of nitrogen. The specified water use for forage is out of the range of the observed data. Table 32 shows the profit maximizing quantities of water and nitrogen given various prices of water with corn grain price of 1.78 cents per pound and a nitrogen price of 10 cents per pound. Table 32. Profit maximizing quantities of water and nitrogen for corn grain with five prices of water (The price of corn grain is 1.78 cents per pound and the price of nitrogen is 10 cents per pound) Price per acre Inches of Pounds of nitrogen inch of water water applied applied $ 0 9.25 93.8 .50 8.79 93.3 1.00 8.33 92.7 1.50 7.85 92.4 2.00 7.39 91.9 The demand curve traced using the data in Table 32 is shown in Figure 24. The curve is slightly more elastic, but because of the very small inter action term in the production function it is almost the same as the static demand curves. If the interaction term was very large the dif ference would be more pronounced. In using this data for deriving demand curves, the static derivation is quite appropriate because of the inde pendence of the effect of the inputs upon output. In the case of a farmer growing corn on the type of soil the Colorado experiment was conducted on, the value of an additional acre inch 156 of water is given by the demand curve if all other inputs are held con stant. For example, if the nitrogen application was 100 pounds, which is about optimal for the prices assumed, and the farmer was applying eight inches of water, he would be willing to pay up to 2 9 cents for another inch of water to put on his corn for that would be the marginal value of the increase in yield that could be obtained by applying one more inch of water. If the total value of the yield increase due to water could be captured, up to 82 cents could be paid for the additional inch of water. Of course, even if the other inputs remained constant in use, their mar ginal value product would increase due to a greater use of water and if 82 cents were paid to water, the other factors could not be rewarded equal to their marginal value product. Summary Budgeting procedures have long been used by economists to estimate the economic value of irrigation water. Given the advent of computers, budgeting for most problems is an inefficient use of scientific manpower. The inability to isolate an optimum organization of farm resources is a major drawback of the budgeting process. The economic imputation of the value of irrigation water by procedures of budgeting usually overstates the worth of water because water is imputed all residual income. Linear programming is a modern refinement of budgeting. Most of the same problems and deficiencies of budgeting apply to linear programming except an optimum can be easily isolated, the computation time is greatly reduced, and a more accurate value of water is given. Linear programming. 157 like budgeting provides answers no better than the data and coefficients which are used. The shadow price of scarce resources indicates the maxi mum amount a farmer would be willing to pay for another unit of the resource. The calculated shadow price is valid only in the neighborhood of the current use. Applying additional units of water, for example, may depress the shadow price to zero rather rapidly. Using linear programming models, the normative demand for water can be derived. From such a demand schedule it can be determined what farmers would be willing to pay for additional water supplies. The classical production function and Euler's theorem provide an easy method of imputing the economic value of irrigation water when there is available a continuous well defined production function. Given the complexity of the response of crops to water as outlined in the previous chapter, the classical production function approach loses some of its use fulness. It is still helpful in analyses involving single irrigation cycles as illustrated previously. Using the estimated production functions from the Colorado experiment, a number of derived demand curves were estimated. The demand curves are static in the sense that all other inputs are assumed constant. A less restrictive curve can be generated by allowing both water and nitrogen to vary simultaneously. The demand for water is very inelastic in that the adjustment in quantity used due to price changes is very slight- Since the state of the arts concerning the response of crops to irrigation water has only recently received adequate attention, very few 158 production response coefficients are available for model formulation. The most adequate method for imputing value to irrigation water would be one which was couched in the framework of dynamic programming. Holding all inputs constant except irrigation water and labor, the optimal irriga tion strategies are ascertained as on farm water supplies varied. The value of any quantity of supplemental irrigation water could then be derived. Such a model could take into account the differences in the value of water applied at different times during the production season. 159 SUMMARY This study was concerned with providing a generalized framework of the production function for crops using irrigation water and consider ing alternative procedures for imputing the economic value of irriga tion water. In order to properly consider the imputation of the economic value of irrigation water, one must have a concept of the relevant pro duction function; to ascertain the correct function a knowledge of water- plant relationships is required; before the water-plant relationships are meaningful in a production function approach, one must understand the nature of the production function concept. The production function or yield response curve is a non-negative mathematical function relating quantities of inputs employed to the quantity of output produced. It presupposes technical efficiency and provides the maximum output obtainable for various combinations of inputs. It is conceivable that there are thousands of inputs which affect crops, some of which are not yet known. These factors which affect production may be divided into four classes; (1) known factors for which the functions are known but which have very high prices relative to productivity; (2) known factors for which the functions are known; (3) known factors for which the functions are unknown; and (4) unknown factors. A production response may exhibit increasing returns, constant re turns, decreasing returns or some combination of the three. The most im portant algebraic forms used in estimating production functions at present are Cobb-Douglas, Spillman, and Quadratic forms. 160 Irrigation water is an important input in the agriculture of arid and semi-arid regions. A farmer has the opportunity to exercise greater control over the water input than most other factors of production. The soil is a storage area for moisture to be used by the plant and the water supply must be replenished periodically. The availability of moisture to the plant is an important factor affecting crop growth and yield. Numerous influences affect the moisture availability and thus the plant response to soil water- The growth of most crops produced under irrigation is stimulated by moderate quantities of soil moisture and re tarded by excessive or deficient amounts. Of great importance to the imputation of the value of irrigation water is the nature of the moisture- plant growth relation. One theory, the "equal facility" school of thought argues that plant growth is substantially the same in the range of optimum moisture- Another theory, the "inverse function" school of thought, argues that the growth rate of crops declines continuously as the moisture con tent of the soil changes from field capacity to the permanent wilting point. The weight of evidence seems to indicate the "inverse function" theory is applicable to the response of most crops. The idea of relating plant growth first to moisture tension and only later to water quantity has been a great step forward in agronomic research with water. The intro duction of the concept of moisture tension makes it possible to construct production functions which are, in a sense, independent of soil type and water quantity. A computer irrigation scheduling model indicates the limited usefulness of accepting the "equal facility" theory. The model is basically a budget which keeps track of the amount of available soil 161 moisture. Based upon the empirical formulations for estimating consumptive use, the model indicates constant water use with widely varying irrigation regimes. Usually, the initial reaction of economists confronted with the problem of determining a water response function in agriculture is to set up a series of experiments in which water quantity is the independent variable and crop output the dependent variable. The problem facing one using the quantity of water approach is that the involvement of time as a variable through the production cycle allows an infinite number of water application patterns. The true response of crops to irrigation water is not known with any degree of certainty. A production response for the season is made up of many sub-period responses which may be quite similar or very different depending upon the crop under consideration. The irrigation problem is made up of a large number of decisions over time, all of which are not independent- The characterization of the production function for specific crops is a dynamic programming problem in which an optimal strategy would be ascertained, given the information about the response of the crop in various sub-periods. Specification could be made up of a large, yet discrete, number of sub-problems which need to be solved to obtain the optimal solution for the crop season. The empirical requirement is to specify those moisture tensions which will be optimum in the sense of maximizing profit to water appli cation. In the simple model presented, maximization of profit per unit of time was shown to be the appropriate criterion for maximizing the profit 162 to water application. The results of an irrigation fertilizer study at Fort Collins, Colorado indicates the response of corn follows traditional economic thinking. The quadratic function was fitted to the data and statistical analysis indicated about 65 percent of the variation in grain yield was explained. About 60 percent of the forage yield variation was explained by the analysis. Presently, there are three direct approaches for estimating water values in irrigation: budgeting or residual imputation, linear programming or activity analysis, and classical production function analysis. The first method, budgeting, is an historically widely used method of factor valuation in which the product value imputed to a given resource is taken as the difference between the total value product and the sum of algebraic products of the quantities of resources multiplied by their market prices. The linear programming approach, the second method, is really systemized budgeting and has the advantage of isolating a single optimum without re sorting to the time consuming trial and error budgeting approach. It allows the marginal value product of each limiting factor of production to be determined simultaneously with the system of equations being solved. The third method used extensively for estimating marginal productivities is based upon derivation of empirical production functions. Euler's theorem provides an easy method of imputing the value of irrigation water when one has a continuous well defined production function. Given the complexity of the response of crops to water the use of the classical production function approach requires modification. It is still very useful in analyses involving single irrigation cycles. Using the estimated produc 163 tion functions from the Colorado experiment, a number of derived demand curves were estimated. The demand curves are static in the sense that all other inputs are assumed constant. A less restrictive curve can be generated by allowing both water and nitrogen to vary simultaneously. The demand for water is very inelastic in that adjustments in quantity used due to price changes is very slight. Since the state of the arts concerning the response of crops to irri gation water has only recently received adequate attention, very few pro duction response coefficients are available for model formulation. The most adequate method for imputing value to irrigation water would be one which was couched in the framework of dynamic programming. Holding all inputs constant except irrigation water and labor, the optimal irrigation strategies are ascertained as water supply varies. The value of any quantity of supplemental irrigation water could then be derived. Such a model could take into account the differences in the value of water applied at different times during the production season. In order to properly value such an important resource as water, con siderable further agronomic research must be conducted. Care should be taken to insure experimental designs which will provide data suitable for estimating the parameters of crop response functions. The response of crops to irrigation water is really a dynamic problem and considerable at tention should be given to the development of suitable dynamic programming routines utilizing available data. Such programs could have "valuation" elements incorporated so that an accurate estimate of the value of addi 164 tional water supplies would be ascertained. Investigation should pro ceed on the feasibility of incorporating simulation programs wherein the actual development of the plant is represented in the computer. The simulation aspect would allow the consideration of the influence of many other factors affecting production such as soil characteristics and weather elements. The effect of the interaction of such exogenous factors upon the optimal irrigation strategies will be very useful in the calcu lation of the value of irrigation water over many crop seasons. 165 SELECTED BIBLIOGRAPHY Auer, Ludwig. Impact of crop-yield technology on U.S. crop production. Unpublished Ph.D. thesis. Ames, Iowa, Library, Iowa State University. 1963. Baumol, William J- Economic theory and operations analysis. 2nd ed. Englewood Cliffs, N.J., Prentice-Hall, Inc. cl965. Haver, L. D. Soil physics. 3rd ed. New York, N.Y., John Wiley and Sons, Inc. 1965. Beringer, Christoph. An economic model for determining the production function for water in agriculture. California Agricultural Experiment Station Research Report 240. 1951. Beringer, Christoph. Some conceptual problems in determining the pro duction function for water. Western Farm Economics Association Conf. Proc. 32: 58-70. 1959. Bishop, C. E. and W. D. Toussaint. Agricultural economic analysis. New York, N.Y., John Wiley and Sons, Inc. cl958. Blaney, H. F. and W. D. Criddle. Determining consumptive use and irriga tion water requirements. United States Department of Agriculture Technical Bulletin 1275. 1962. Briggs, L. T. and H. L. Shantz. The water requirements of plants. United States Department of Agriculture, Bureau of Plant Industry Bulletin 284. 1913. Buckman, Harry 0. and Nyle C. Brady. The nature and properties of soils. 4th ed. New York, N.Y., The Macmillan Co. 1963. Cesal, L. C- Normative resource demand functions and elasticities pro grammed for representative farms. Unpublished Ph.D. thesis. Ames, Iowa, Library, Iowa State University. 1966. Clark, Colin. The economics of irrigation. New York, N.Y., Pergamon Press. 1967. Clark, J. B. The distribution of wealth. New York, N.Y., The Macmillan Company. 1899. Fellows, Irving F-, editor. Budgeting: tool of research and extension in agricultural economics. Connecticut (Storrs) Agricultural Experiment Station Bulletin 357. 1960. 166 Ferguson, C. E. Microeconomic theory. Homewood, Illinois, Richard D. Irwin, Inc. 1966. Friedman, Milton. Price theory - a provisional text. Chicago, Illinois, Aldine Publishing Co. 1962. Golz'e, Alfred R. Reclamation in the United States. Caldwell, Idaho, The Claxton Printers, Ltd. 1961. Goony, J. J. and J. E. Pehrson. Avocado irrigation. University of California Extension Leaflet 50, • 1955. Hagan, Robert M. Water-soil-plant relations. California Agriculture 10, No. U: 9-12. April, 1957. Heady, Earl 0. Economics of agricultural production and resource use. Englewood Cliffs, N.J., Prentice-Hall, Inc. cl952. Heady, E. 0. and Wilfred Candler. Linear programming methods. Ames, Iowa, Iowa State University Press. cl958. Heady, E. 0. and John L. Dillon. Agricultural production functions. Ames, Iowa, Iowa State University Press. cl961. Heady, E. 0. and Luther G. Tweeten. Resource demand and structure of the agricultural industry. Ames, Iowa, Iowa State University Press. cl963. Henderson, James M. and Richard E. Quant. Microeconomic theory. New York, N.Y., McGraw-Hill Book Co., Inc. 1958. Hendrickson, A. H. and F. J. Veihmeyer. Irrigation experiments with grapes. California (Berkeley) Agricultural Experiment Station Bulletin 728. 1950. Israelson, Orson W- Irrigation principles and practices. New York, N.Y., John Wiley and Sons, Inc. cl956. Jensen, M. E. and H. R. Haize. Estimating évapotranspiration from solar radiation. Journal of Irrigation and Drainage 89: 15-41. 1963. Lowry, R. L. and A. F. Johnson. Consumptive use of water for agriculture. American Society of Civil Engineers Conference Proceedings 67: 595-616. 1941. Miller, Stanley Frank. An investigation of alternative methods of valuing irrigation water. Unpublished Ph.D. thesis. Corvallis, Oregon, Library, Oregon State University. 1965. 167 Miller, Stanley Frank and Larry L. Boersma. Economic analysis of water, nitrogen, and seeding rate relationships in corn production on Woodburn soils. Oregon Agricultural Experiment Station Technical Bulletin 98. 1966. Moore, Charles V. A general analytical framework for estimating the production function for crops using irrigation water. Journal of Farm Economics 43: 876-888. 1961. Penman, H. L. Estimating evaporation. Transactions of the American Geophysical Union 37: 43-46. 1956. Renshaw, E. F. Value of an acre foot of water. Journal of American Water Works Association 50: 303-309. 1.958. Richards, L. A. and C. H. Wadleigh. Soil water and plant growth. Agronomy 2: 73-251. 1952. Russell, Sir E. John. Soil conditions and plant growth. 8th ed. New York, N.Y., Longmans, Green and Co. 1950. Shaw, Byron T., ed. Soil physical conditions and plant growth. New York, N.Y., Academic Press, Inc. 1952. Taylor, S. A. Estimating the integrated soil moisture tension in the root zone of growing crops. Soil Science 73: 331-340. 1952. Thome, D. W. and H. B. Peterson. Irrigated soils, their fertility and management. Philadelphia, Pa., The Blakiston Company. 1949. Thornthwaite, C. W. An approach toward a rational classification of climate. Geographical Review 38: 55-94. 1948. Visser, W. C. Crop growth and the availability of moisture. Institute for Land and Water Management Research, Wageningen, Netherlands, Technical Bulletin 6. 1959- Wadleigh, C. H. The integrated moisture stress upon a root system in a large container of saline soil. Soil Science 61: 225-238. 1946. Wadleigh, C. H. and A. D. Ayers. Growth and biochemical composition of bean plants as conditioned by soil moisture tension and salt concentra tion. Plant Physiology 20: 106-132. 1945. Wadleigh, C. H. and H. G. Gauch. Rate of leaf elongation as affected by the intensity of the total soil moisture stress.; Plant Physiology 23: 485-495. 1948. 168 Wadleigh, C. H., H. G. Gauch, and 0. C. Magistad. Growth and rubber accumulation in guayule as conditioned by soil salinity and irrigation regime. United States Department of Agriculture Technical Bulletin 925. 1946. Yaron, Dan. Resource allocation for dairy and field crops in the Negev area of Israel. Unpublished Ph.D. thesis. Ames, Iowa, Library, Iowa State University. 1960. 169 ACKNOWLEDGMENTS The author wishes to acknowledge the guidance, patience, encourage ment and helpful suggestions of Dr. Earl 0. Heady throughout this study. Thanks is extended to the Bureau of Reclamation at Denver, Colorado which furnished pertinent data, financial support and cooperated fully to make this study possible. Appreciation is especially given to Ira Watson and Robert Struthers of the Bureau for their suggestions and en couragement. To Mary, his wife, who has rendered support through the joys and sorrows which accompanied this study goes the author's love and grati tude. To Kathy, Lynda and Robert; thanks for making life the joy that it is. The author extends sincere appreciation to his wife and children for their inspiration, encouragement and love without which any worldly achievement would be empty. The author is indebted to Him who is the greatest scientist of all and who allows men to make scratches at the infinite. 170 APPENDIX Only the more interesting computations are shown of the irrigation scheduling model. The actual computer print-out contained 22 columns of information whereas only nine of the more important are given here. The model computed values for 228 days with five extra days on each end to allow for forward projections. The column "Day Growth" specifies the month and the day, for example, 5.04 is May 4th. The "Act Ave Temp" is the actual recorded mean daily temperature in degrees as the season progresses. This is the main input from which the actual consumptive use is calculated. "Cons Use U" is the calculated consumptive use of moisture by the crop each day through the season in inches. The figures are the same re gardless of the irrigation criterion used. Only the 100 percent criterion shows the actual consumptive use figures. The summation of the daily consumptive use since the last irrigation is given in the column headed "Accum U". The units are inches. Historical climatic data is used to project what the consumptive use will be over the next five days. An adjustment is made for current ab normally high or low temperatures. The column "Adj Et 5 Fwd" is an esti mate of soil moisture use in inches for the next five days. "Max Avl H20" is the amount of available inches of water the root zone could hold if at field capacity. As the season progresses, the root zone increases until some maximum is reached. 171 The actual inches of moisture available to the plant on a specific date is given in "Act Avl H20". Adding what moisture is now available to any new moisture supplies that might be encountered as the root zone increases gives the total inches of available moisture for the next five days. This sum is recorded under "Tot Avl H20". The percentage of available moisture remaining in the soil is calcu lated from the total available and what the maximum available could be. ' This result is given in the column headed "Surplus H20". The maximum is lUO percent. The season begins with 100 percent of available moisture in the soil. When the critical value is reached such as 50 percent, the "Surplus H20" number for that day is replaced by 1000. Thus, looking down the column it is easy to pick out the dates of the irrigations. When 1000 is placed in the last column, an irrigation is assumed to occur and calculations proceed accordingly. The results of four different criterion are given for comparative purposes. The four are: (1) irrigate when 100 percent of available moisture has been depleted; (2) irrigate when 75 percent of available moisture is depleted; (3) irrigate when 50 percent of available moisture is depleted; and (4) irrigate when only 25 percent of the available moisture has been depleted. 172 Irrigation scheduling model results using criterion I Irrigate when 100 percent of the available soil moisture is de pleted. 173 l»< 1 ri ACT 4V,- T f ^ C'l.NS 1.: c. F M AC eu M l) c.CCO '•4 0 0.0 C C 4.rrr ss oc: 0.0 0 n A, 0 ]0 0 c. 0 r P.orc 70 400 O.O 0 l.Cro 7'^ 200 0.0 0 4.0 10 71 SCO Û.GUaOH 0 cc° 4.070 COO 1 0.00 ri 2'5. 0 o In 4.0"'^ 5"? 00- 0. CC7?3 0 0?4 4.040 4 1 nro 0.00nl3 0 C?7 n 4.C5^ AS I C 0. r0Q:0 041 r 4.0f0 A'l /. n o 0.01026 05? 4.070 64 Î r 0 C. 0 10^7 ^ n 4.0"0 70 iro 0.01211 0 74 4 .0^0 74 B p '• G.C13 /Q P. H - 4. irn 7 •> ~00 0.0 1 4 1 0 102 4.110 6^ 100 O.Ol i;i4 r 1 Ir r\ 4.1 ?r 60 ^0'-^ . 0. 012 )6 1 P': p. 4. 1 TC 57 20<" O.CL^!o 14? 4.140 AO 400 0.014 15 P 15f; 4.1 • 41 ooo 0.0 14 W 17 1 4.1/0 6 1 ^O" 0. Cl 560 r 1 8 7 4.170 70 4 0^ 0. 318r.8 0 205 4. 1 -0 7? 3C0 0.0/0^3 225 4.ion 74 60O 0.02 U 3 0 247 4.pro 70 OOO 0.0 7C fy 0 267 4.?10 6". o 0 o G. 02 100 C 2GC 4. 270 61 400 G .01'96 1 708 r 4.2^0 6« 60 0. j. 0'2 2 6 3 31 4.240 6^ 20 <"> 0.02 34 7 0 >64 4.2=0 6 A %0o 0.02 41)3 0 -373 4 .?f 0 7"^ Î on 0.02u)2 " 406 4.270 70 per 0. -J .'< 11 "' 4 ^ A, p 4.2M0 79 iQr Ù.030 22 46 7 600 4. 290 ?•-! 000 0.0-4 if 3 4. l^O A4 f'OO' 0. 03 8 )5 0 SIQ 5.010 35 n n •"! 0 .Of OL 2 679 n 5.020 7: 1 Oo 0. 0 64V7 h 13 5 .0 3 0 7" 70 u.0 3 6 ^6 0 ^-50 p, 5.040 70 opr. 0.03 DM8 687 S.OCT 7S 00 "" 0.04 17 5 0 7 29 p 5.060 7n 400 0.04325 772 1 5.0 7C 9 0 r; c r- 0.04 738 9 1^ r 5 .OPO R6 600 0.05 334 871 5. OCC PO 1 on C. C5S32 0 -31 5. IOC ^0 SCO 0 .Ot. 137 0 OÇ? 050 5.110 92 i 0 0. 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"""r - !1 .1 1 ro ::o 1 1. 1 z"" 1 r\ 1 '"r. rnr 11.1 1CC-r-r r 11.1"^ r, 1rc.rrr 188 Irrigation scheduling model results using criterion II Irrigate when 75 percent of the available soil moisture is de pleted- 189 GHJm r H ACT AVc rCYP .LiMS LSC U ;cjM u -5. or e 64.800 0. & 0. .JCO 63.400 0. 0 c -3.000 69. 10 0 0 .0 0 .0 -2.CL0 70.400 0. 0 1.0 -i . 3:ic 7 3.2Gn 0.0 0.0 4.01C 71. 800 0. 0 0 .004 4.320 6d.00C o.c 0. 016 i. 0 30 59.000 0 .0 .) .0 24 4.040 ol.000 0. 0 ). 032 +, U). 10 L G .0 0.04 1 4.060 6^1.400 0.0 0. 05? 4. 0 fC 64.300 0 .0 0 .0 6 2 4.J HO 70.100 0. 0 ). 074 -f .0 -)0 7+.rfOO 0.0 0. 4. LOO 7 200 0. 0 ].in; ^.110 64. 100 0.0 0. 1 lo 4. 12 0 6 0.300 0.0 D .1 29 4. 130 57.200 0. 0 0. 142 4. 140 UJ.4C0 0.0 ]. 15o 4. 150 61. 000 0. 0 ).171 1. 160 ol.300 0 .0 0. 18 7 c o 4. i7C 0. 0 3.205 4.140 72.300 o.c 0.225 4. 140 74.600 0.0 0.?47 4.200 7:1.000 0 .0 0. 267 t. 210 6 -s. 000' 0.0 0 .289 t .220 61.400 0. 0 0. 30 3 T. 3 31 t.230 o3 « oO ;) 0 .0 4. 240 OH.20 0 0. c 1. 354. 4.2^0 63.300 0.0 0. 374 4 .2iS0 7 3.100 0. 0 0.405 4.2^0 fj.2Gr 0.0 ).436 H. 280 7a.300 0.0 0.467 4.290 79.900 0.0 0. 500 4. 300 3 +. t>00 o.c 0 .5:39 5 .310 {33. CGC 0. 0 0. 5 79 5.020 70 .100 0 .f) 0 .61 3 5.030 73.200 0.0 0.650 5 *040 7J.900 0.0 0.687 5.Cb0 7b. 800 0.0 0 .729 3. 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').926 S3.21 3 10.240 b. t58 57.678 10.290 5. ?9 3 57.204 IG. 3CC 5. 73 0 56.763 10. 5 10 6.676 56. 373 11.010 ;>.627 56.064 il .020 5.578 55.781 11.030 5.534 55 .487 11.040 5.492 55. 210 11.050 5 .45 1 54.9 36 11.060 5. 408 54.6 06 11 .070 3. 368 54.232 11. CdU 5.327 53.865 11.JVO 5.237 5 3. 475 11.100 5.248 53 .0 3 7 11.110 5. 208 52.692 11. 120 5.169 52 .309 11.130 5.133 51. 9o0 i 1. i + 0 5 .099 51.660 11.150 0. 0 100.000 11.160 0. ) 100.000 11. 170 0.0 100.000 11.180 0. 0 100. 000 11.1^0 U * 'J 100 .000 204 Irrigation scheduling model results using criterion III Irrigate when 50 percent of the available soil moisture is de pleted. n > < en •» I I I 1 I c; vr vT vn in m vn m ^ f-» M vr, s; • *•••••••* •*•••••••• ••••••# #•• •••••••••• —< (— i_>-'i_i,-4(-<0000 O o o o o r.) 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