a test of a linear programming model of agriculture
TRANSCRIPT
A TEST OF A LINEAR P R O G R A M M I N G
MODEL OF AGRICULTURE
RICHARD HOWES, Richard Howes and Associates
This paper reports a phase of a continuing effort in the United States Department of Agriculture to develop better methods for evaluating water resource investments.
Current procedures employed by federal agencies in planning water resource development require that economic studies be conducted in order to measure the benefits and costs resulting from a particular set of investments. 1'2 However, the interrelationship between the prices of water-related goods and services and quanti ty of water available has limited the usefulness of these studies. Typically, values of goods and services resulting from water resource invest- ments are utilized to establish a measure of benefits attributable to a project. However, water resource development results in shifts in the supply functions for those goods and services. In many cases, the prices of goods and services available for planning water resource investments will, therefore, be higher than those which will be in effect after the installation of the project. This is particularly true in cases where water development goods and services are supplied to relatively small markets.
Economic feasibility studies of water resource projects have been criticized because they failed to (1) take into account alternative methods of producing the goods and services attributed to water resource investments; (2)describe the impacts of water development upon persons and institutions not directly affected; and (3) measure the values of water necessary for efficient allocation among competing uses. 8 An appropriate mathematical programming model could provide insights into these problems. However, the utilization of mathe- matical models in planning water development projects has not been widespread. In this study, an interregional linear programming model was tested in an effort to develop a model which could be used to specify economically feasible water resource investments in the Susquehanna River Basin. The use of a model of
1 Policies now in effect are summarized in Policies, Standards and Procedures in the Formulation, Evaluation and Review of Plans for Use and Development of Water and Related Land Resources, The President's Water Resources Council, May 29, 1962.
2 For a more complete treatment of the concepts behind the policies see 0. Eckstein, Water Resource Development (Cambridge, Massachusetts: Har~card University Press, 1961) and Proposed Practices for Economic Analysis of River Basin Projects, Report to the Interagency Committee on Evaluation Standards, May, 1958.
A number of recent studies were designed to improve these analytical methods; see, for example, G.S. Tolley and F. E. Riggs, Economics of Watershed Planning (Ames, Iowa: Iowa State University Press, 1961). A. Maass, et. al., Design of Water Resource Systems (Cambridge, Massachusetts: Harvard University Press, 1962). R.N. McKean, E~ciency in Government through Systems Analysis (Chicago Illinois: John Witey, 1958).
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this type could provide simultaneous est imates of benefits resulting f rom a project and marke t prices, thus cutt ing through the circularity described above. It could also provide a technique by which the min imum benefit-cost ratio could be considered as an additional constraining factor, as suggested by Eckstein.
The model could provide a working tool to be used throughout the planning process. Solutions of the model under al ternative assumptions could be made, and partial equilibrium analyses carried out by utilizing the resulting data. Fur thermore , if such a model were developed, optimal solutions could be computed with and without the inclusion of water development alternatives, and, through the use of this technique, changes in important economic variables could be measured. The effect of these changes on resource owners and consumers could be estimated. This information would contribute to a fuller understanding of the secondary or indirect effects of water development.
In short, such a model, if it could be developed, would offer substantial advantages over al ternative methods of economic evaluation of water resource investments. However, before such a model can achieve its m a x i m u m use- fulness, its ability to simulate the phenomena in question must be tested.
Linear p rogramming can be thought of as a system for determining a set of logical consequences from a set of specific assumptions. In economic studies utilizing linear programming, some of the assumptions of the model relate to physical and economic conditions making up the economic environment in which a hypothetical economic unit finds itself. It is sometimes impossible to make direct tests to determine whether or not these assumptions approximate actual economic behavior. However, it is sometimes possible to test the as- sumptions of the model indirectly, using, as a basis for comparison, the set of logical consequences resulting f rom the assumptions. Through this method, it may be possible to develop sets of assumptions universally applicable to the type of economic units being studied.
In this study, a test was made of the applicability of a specific set of assumptions regarding the use of natural resources in agriculture in the Susquehanna River Basin. An interregional linear p rogramming model was developed from this set of assumptions. Production and resource use specified by the model were compared to actual production and resource use. If these quantities were in close agreement , it could be concluded that the assumptions of the model adequately simulated the behavior of economic units. These behavioral aspects found in one situation might be assumed to be generally t rue for the type of economic units studied and could be used in models to predict the responses of these units to changes in the physical and economic environment. In this specific case, if the model could adequately simulate existing use of natural resources, it would be concluded that the set of assumptions regarding the behavior of economic units was appropriate for use in a study to predict changes in production and resource use associated with water development for agriculture.
HOWES: TEST OF A LINEAR PROGRAMMING MODEL 125
MODEL
T h e cons tan t s of the mode l are:
(c = 1 . . . . . n) r~afo (e = 1 . . . . . m)
(L = 1 . . . . . U)
L (c = 1 . . . . . n) ~aia (d = 1 . . . . , k)
(L = 1 . . . . . U)
L (c = 1 . . . . . n) E ~ (L = 1 . . . . . U)
'dafb,~qb~b ( d , q = 1 . . . . . k) ( b = 1 . . . . , h )
i,~bfb, iqbsb (d,(b = ~,,. �9 �9 q 1, ",'Izi k)
O f , . J - - 1 . . . . . U + H )
J (b = 1 . . . . . h) Oib ( j = 1 . . . . . U -t- H )
L (c = 1 . . . . . n) r~C~ (e = 1 . . . . m)
(L = 1 . . . . . U)
L (c = 0 . . . . . n) ~Cid (d = 1 . . . . . k)
(L = 1 . . . . . U)
L J ( e = l . . . . , m ) ts. (L = 1 . . . . . U)
( J = 1 . . . . . g + H )
L J ( b = l . . . . . h) t& ( L = 1 . . . . . U )
(J = 1 . . . . . U + H) L J ( b = l . . . . . h)
~dCsb (C 0 . . . . n) ~c (d = 1 . . . . . k)
(L = 1 . . . . . U) ( J = 1 . . . . . U + H )
LJ ( d = l . . . . . k) t~d (L = 1 . . . . . U)
( J = 1 . . . . . U + H )
L (L = 1 . . . . . U) Bib (b = 1 . . . . . h)
T h e so lu t ion v a r i a b l e s L J ( c = l . . . . . n)
roSf~ (e 1 . . . . m) (L = 1 . . . . . U) ( J = 1 . . . . . U + H )
A c r e s of land of t ype c r equ i r ed to p roduce one uni t of final c rop e in r eg ion L.
A c r e s of land of t y p e c r equ i r ed to p roduce one un i t of i n t e r m e d i a t e c rop d in reg ion L.
Acres of land of t y p e c ava i l ab l e for a g r i c u l t u r a l use in reg ion L.
I n t e r m e d i a t e p roduc t s d and q, r e spec t ive ly , r equ i r ed per un i t ou tpu t of final c o m m o d i t y b. T h e s e coef- f icients a re cons t an t a m o n g regions .
T h e l imi t of the p ropor t ion of i n t e r m e d i a t e p roduc t s ix and iq, r e spec t ive ly , wh ich can be used in the p roduc t ion of final good b. T h e s e coefficients a re cons t an t a m o n g regions .
T h e q u a n t i t y of final good e wh ich m u s t be supp l i ed to reg ion J.
T h e q u a n t i t y of final good b wh ich m u s t be supp l i ed to reg ion J .
T h e cost of p roduc ing final crop e on soil t ype c in r eg ion L.
T h e cost pe r uni t of p roduc ing i n t e r m e d i a t e c rop d on soil t y p e c in r eg ion L. T h e des igna t ion c = 0 m e a n s t ha t the i n t e r m e d i a t e feed g ra in was i m p o r t e d r a t h e r than g r o w n w i t h i n the region .
T h e t r a n s p o r t cost pe r un i t of final good e sh ipped f rom reg ion L to reg ion J.
T h e t r a n s p o r t cost pe r uni t of final good b sh ipped f rom reg ion L to reg ion J.
Cost in r eg ion L of p roduc ing final l ives tock p roduc t b, us ing i n t e r m e d i a t e d g r o w n on soil t ype c, and sh ipp ing i t to reg ion J.
Costs of t r a n s p o r t i n g one uni t of i n t e r m e d i a t e d f rom reg ion L to reg ion J.
Costs of c o n v e r t i n g i n t e r m e d i a t e s into one uni t of final good b. I t is a s sume d tha t no d i f ferences ex i s t a m o n g feeds in the i r convers ion costs pe r un i t of fb.
of the p r i m a l p rob l e m a re specif ied as:
S h i p m e n t s of final good e p roduced on land of t y p e c in r eg ion L and sh ipped to reg ion J .
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J (c = 0 . . . . . n) Shipments from region L to J of final good b idS1b (d = 1 . . . . . k) produced from intermediate d, which was in turn *~ (b = 1 . . . . . h) grown on land of type c.
(L = 1 . . . . . U) ( ] = 1 . . . . . U + H)
It is assumed that there are two processes that can be used in production of final agricultural commodities. One is the process of producing final crops that move directly from the agricultural sector. The cost of a unit shipped from region L to ] of final crop e produced on soil type c is
L J L L J
%Cf e = %Cf~ + t fe . (1)
The second is a process of combining feed crops (intermediates) in the production of livestock products. The cost of shipping final livestock product b is defined in terms of the costs of the respective feed crops used in produc- tion plus costs of transportation and feed conversion.
L J J J L L L J
,~CSb = ~a/b ~, Cid + ~dafb t~d + Bfb + tfd �9 (2)
In (2), interregional shipments of intermediate feed crops are permitted. However, forage crops are substantially heavier than the livestock products into which they can be converted. It is, therefore, assumed that forage crops will not move in interregional trade. The ability of a region's resources to produce forage crops will, therefore; limit its production of livestock products.
However, purchased feed grains can be substituted for locally grown grains or forage in the production of livestock products. Prices of feed grains have been shown to vary regularly with distance from the major grain producing region of the United StatesJ Therefore, given prices of feed grains in the major surplus area, prices in other regions can be calculated. The following formulation includes the alternative of producing livestock products using purchased feed inputs, excluding interregional shipments of forage from consideration.
L ~ J L L L ~ J
i d C f b ~--- ~dafb r. Cid -[- Bfb ~- tYb (3) r c
L
where r0 C~e is the market price of feed grain ix purchased and used in the production of fb.
The objective function is to minimize the total cost of producing and shipping final crops and livestock commodities from the region to their respective market areas. 5
4 G.G. Judge and T.A. Hieronymus, "Interregional Analysis of the Corn Sector," (Urbana, Illinois: Department of Agricultural Economics, University of Illinois, 1962), p. 14.
5 The equivalance of this model and the maximum net returns model for all farms combined was shown in A.C. Egbert and E.O. Heady, Regional Adjus tments in Grain Production, Supplement to Technical Bulletin 1241, (Washington, D.C.: United States Department of Agriculture, 1961) p. !0,
HOWES: TEST OF A LINEAR PROGRAMMING MODEL ] 2 7
Minimize
Z = ~ : ~: [ ~'+'L ~--~ ~ 5. (,oC:, + t:.)~oS:. c=l e= l L : I J : l
f L ~ J L ~ J k h U U + H L L
+ 2 2 2 2 (~as~,.oc~,, + Bs~ + tsp4Ss~. e : 0 d : l b ~ l Z : l J = l
(4)
In this formulation, the final livestock product is specified as produced with a specific feed crop which, in turn, is grown upon a specific type of soil. A
L ~ J
more usual presentation would be to specify the solution variable as Sib. However, this scheme would have specified the opt imum pat tern of shipments of livestock products without specifying either the s t ructure of intermediate goods used in production or the result ing land use pattern.
The constraints of the problem follow. First, land resource of type c used in the production of intermediate and final crops cannot exceed the amount of this type land available for agriculture in region L. We can establish the following relationship because of the assumptions: ( a ) the re are no interregional shipments of forage, and (b) purchased feeds are imported.
m U § L L J k L L ~ L L
Z Z ro~f~ S i , + Y. .r2id S~ a <_ E,~ (5) e = l J = l d = l
L
where roa~ a = 0 .
The production of intermediate d for use in the production of livestock products within region L can be stated as
L ~ L h U ~ I I L ~ J
S~,~ = ~. ~. id~fb S f b . (6) b = l J = l
L L
By substi tut ing for S~ d in (5) and mult iplying by - 1 the first set of constraining relationships is obtained.
m U + H L L ~ J h k U + H L I ~ J L
- Z 5 roaf~ Sf~ - ~. ~. ~. r2 ig id~fb S f b >- -- E~, �9 (7) e = l J = I b = l d = l J = l
L ~ J
Each shipment Sfb can be thought of as the result of a t ransformation of a specific feed crop, grown on a specific soil type.
L L *
let roala ia~fb = r~d~fb �9 (8)
Equation (7) can be rewri t ten as
m U + H L L ~ J h k U + H L * L ~ J L
e = l g= l b = l d = l d = l
A constraint was next introduced that limits the ratio in which products ia and iq can be combined in the production of final good lb.
U + H L ~ J U - - H L ~ J
iabfb ia~fb ~. S f b -- iqbfb iq~fb ~. Sf~ >__ O. (10) J = l J : 1
128 PAPERS OF THE REGIONAL SCIENCE ASSOCIATION
This statement is equivalent to requiring that
Uq-H L ~ J z s , i.% ~=0 i,~ b > (11)
U+H L ~ J
Following the notation employed above, (10) can be reduced, since the shipments of livestock products are specified as from a given intermediate grown on land of a particular soil type.
n U~-H L ~ J n U-kH L ~ J
- b,oo o >_ o (12)
From a conceptual standpoint, (10) is preferable to (12), since in the former intermediates are combined in a specific way in the production of fb. In (12) it appears that only i~ is used in the production of some fb, and only iq is used in the production of another increment of fb. However, these expressions are equivalent as shown above.
A set of constraints which have the form of (12) permit flexibility in the production function for specific livestock products. For example, a constraint of this type could be employed to limit the proportion of roughage in a ration fed to dairy animals. The source of the roughage could be silage, hay, and pasture. A similar constraint could be used to limit the proportion of grains fed to dairy cattle, with the grains being imported or locally produced.
A further set of constraints specified that shipments of final goods to each region must be at least equal to a minimum level.
f U L ~ J J %S~ _> Ozo �9 (13)
c : l Z = l
f h U L ~ J J Z ~. ,idSsb >-- 016 �9 (14)
c = l d = i L = I c
Finally, it is required that all shipments be nonnegative. 6
L ~ J L ~ J %S~, ~S~ b _> 0 . (15)
e
One of the assumptions of the model was that the region considered is closed with respect to demand for its agricultural output. On the other hand, production of livestock products could result from use of locally grown or imported feed grains, and, therefore, it is an open region. However, it is not unusual for this situation to exist because of differences in transport costs between feed grains and livestock products. It is especially true of milk
s This model, which considers the efficiency of the agricultural economy as a whole without specific reference to the individual firm, is similar to models used in A.C. Egbert, E.O. Heady, and R.F. Brokken, "Regional Changes in Grain Production: An Application of Spatial Linear Programming," Research Bulletin 521, (Ames: Iowa Agricultural Ex- periment Station, January, 1964); and J.M. Henderson, "A Short-Run Model for the Coal Industry," l~eview of Economics and Statistics, XXXV!.I (blovernber, 1955), pp. 336-46.
HOWES: TEST OF A LINEAR PROGRAMMING MODEL ] 2 9
produced for use as fresh milk and cream. Regional analyses of the milk industry and the feed grain sector of
agriculture have been made using linear programming. The dairy study showed that at equilibrium the northeastern United States would produce the fluid milk that it required. 7 In contrast, the study of the corn sector of the United States showed that the midwest would ship feed grains to all other regions of the country. ~ In this case, therefore, the northeast could be con- sidered a closed region with respect to milk production and an open region with respect to feed grains.
In an area which specializes in the production of livestock commodities, consumer demand exists for the output of final commodities. The demand for feed grains is derived from demand for final goods and can be supplied from local production of grains and roughage crops or from imports of feed grains.
D A T A
The linear programming model outlined above was employed in a study of the agricultural economy of the Susquehanna River Basin, Figure 1. Four agricultural production regions within the basin were delineated. Seven regions, where the agricultural products of the basin are consumed, were delineated. Four of these regions were within the basin and three were outside: New York-Northern New Jersey, Philadelphia-Southern New Jersey-Wilmington, and Baltimore-Washington, D.C.
Estimates of milk production and milk supplied to each consumption region in 1963 were made, based upon published data. Estimates of the production of other agricultural commodities in 1963 were made for the basin as a whole. These estimates were then used as production and shipments targets in the linear programming model.
The conservation needs inventory provided estimates of the quantities of agricultural land of various types found in each production region. 9 Each production region was found to have between forty and seventy types of agricultural land for which differences in agricultural productivity could be determined. This inventory of land resources and a survey conducted in a sample of counties in the basin were used to estimate the cropping pattern in 1963 for each soil type. The cropping pattern in 1963 provided one base to which the results of the model were compared.
Crop yields per acre according to soil types were obtained from the Pennsylvania office of Soil Conservation Service, United States Department of Agriculture. Estimates of the average cost of producing individual crops on each soil type were based upon engineering standards and these estimates of
7 M.M. Snodgrass and C.E. French, "Linear Programming Approach to the Study of Interregional Competition in Dairying," Bulletin No. 637, (Lafayette: Indiana Agricultural Experiment Station, 1958), p. 16.
8 Judge and Hieronymus, op. c/t., p. 27. New York Conservation Needs Committee, New York State Soil and Water Con-
servation Needs Inventory, 1963; and Pennsylvania Conservation Needs Committee, Pen- nsylvania Soil and Water Conservation Needs Inventory, n.d.
130 PAPERS OF THE REGIONAL SCIENCE A S S O C I A T I O N
New York
Pennsylvania
REGION II
Pennsylvania
Maryland /
\ \
0ameron~ ] r , ~ :
I Clearfield / Centre~---~-2/ . ~x '~ur ~ J . [ I I I .iz Uni~ ."J-"
/ / ~i~ ~ n v d e r ./umber; ~21 . . . . ~ , J " ~ ' f ~ Z 2 . . . . ./u.a ./'_ / / " - ~ ~ - a ' ~ ' S ~ b u y l k ~ l l
i / ~ " - " 2 - ~ I / ~ " . ~ % . . - ' - ' ~ . . . . . "...,.-~ / ~ ~ f ~ ] Da.u- d- "%,,
) Bia,r .ly_~_rt:',..<" ~:,.<~ " phme<. . . . . .
R ,0N I
I
Cortlandl Chenango [ Otsego .
t w | I - - - - - - I Steuben Chemung] Tioga Broome - - ' ]
, i ! - I I I
,,2 ga 1 Bradford Susquehanna
-- _ ~ _ ~-'~ "/Wyo, .,rag/ ~ Sullivan [ / Lackawanna
\\ Lyeoming ~ g u z e r n e ~\ )
REGION III
r / Lancaster ' / ' ~ f 9 ~'',, (
/Ful- # F r a n k h ~ ~ YOrk xx \, / / ton I dams ,
REGION IV
FIGURE 1. AGRICULTURAL PRODUCTION REGIONS OF THE SUSQUEHANNA RIVER BASIN
average crop yields per acre. Transport costs for milk were estimated at 1.2 to 1.4 cents per hundred
weight per ten miles from studies made in connection with milk pricing in New York, New Jersey, and Pennsylvania. Transport costs for beef, wheat, fruits, and vegetables were not considered because of lack of information concerning the actual places where these commodities were shipped.
The above data were put into a form required by the model, and it was computed to see how closely the model's results agreed with the existing pattern of production and land resource use. Approximately 8,000 activities were included in the model which contained 211 constraining relationships.
COMPARISON
There are many facets to the optimal solution of the linear programming model. These facets are identical to the economic data generated by the
HOWES: TEST OF A LINEAR P R O G R A M M I N G MODEL "[ 3 ]
market mechanism. Furthermore, in all optimal linear programming solutions, the data are the same as would result from a perfectly competitive market in equilibrium. 1~
In the optimal solution, as in a perfectly competitive economy in equil- ibrium, supply and demand are equal for each commodity. In the model, the demand schedule for each final agricultural product is completely inelastic at a specified level. The supply function for each commodity is a linear step function that shows an upward trend with increases in the quantity produced.
Substantial changes in the existing pattern of milk production were specified by the minimum cost linear programming model. Increases in output were indicated for Regions III and IV, and decreases were specified for Regions I and II. Under the optimal pattern for agriculture in the Susquehanna River Basin, each region in the basin was found to be self-sufficient in milk produc- tion. This is due to the relatively high fixed cost of milk transportation when compared to the opportunity costs for the production of feed crops. The production pattern specified by the model for milk in this case is consistent with empirical findings. A large proportion of the milk supply needed for consumption within a region is produced on farms within that region, although some interregional shipments occur.
The optimal solution of the linear programming model indicated that farms in Region III would supply 51.9 per cent of all milk produced within the basin, Table 1. In 1963, Region III produced only 27 per cent of the milk supplied from the basin. The model indicated that Region IV, which was estimated to produce 15 per cent of the basin's milk supply in 1963, would produce 32.4 per cent of the total milk supply.
TABLE 1 PRODUCTION OF MILK AND PERCENTAGE DISTRIBUTION IN THE
SUSQUEHANNA RIVER BASIN, BY REGIONS, ACTUAL AND SPECIFIED BY LINEAR PROGRAMMING MODEL
Region
Region I Region II Region III Region IV
Total
Actual, 1963
Quantity
Million pounds 3,183
214 1,582
876
5,855
Distribution
Per cent 54.3 3.7
27.0 15.0
100.0
From Linear Programming Model
Quantity
Million pounds 841 78
3,037 1,899
5,855
Distribution
Per cent 14.4 1.3
51.9 32.4
100.0
In 1963, Region I supplied approximately 54.3 per cent of the milk produced in the Susquehanna River Basin. However, the optimal solution of
10 R. Dorfman, P. Samuelson, and R. Solow, Linear Programming and Economic Analysis (New York: McGraw-Hill, 1958) pp. 39-129.
] 32 PAPERS OF THE REGIONAL SCIENCE ASSOCIATION
the linear programming model indicated that this region would produce only 14.4 per cent of the milk. According to the model, Region II would produce only 1.3 per cent of the milk supplied by farms in the basin. The supplies originating in Region II were completely utilized for milk consumption within that region.
The optimal solution of the linear programming model specified that Regions I, III, and IV would produce milk required in the New York-Northern New Jersey milk marketing area, Table 2. According to the model, Region III would supply 68.2 per cent of the milk and Regions I and IV, 11.7 and 20.1 per cent, respectively. The New York-Northern New Jersey milk marketing area provided the only export markets for milk produced in Regions I and III, whereas Region IV shipped milk to Regions VI and VII, in addition to Region V. All the milk needed in the Delaware Valley area, Region VI, was supplied by Region IV.
TABLE 2 DISTRIBUTIONS OF MILK SHIPMENTS FROM REGIONS WITHIN THE SUSQUEHANNA
RIVER BASIN TO METROPOLITAN AREAS OUTSIDE THE BASIN, ACTUAL
AND SPECIFIED BY LINEAR PROGRAMMING MODEL
Region of Destination
New York-Northern Upper Chesapeake New Jersey Delaware Vally Bay-Washington, D.C.
Region From Linear From Linear From Linear
Actual Program- Actual Program- Actual Program- ming Model ruing Model ruing Model
Per cent
Region I 71.1 11.7 5.7 -- -- - - Region II 3.4 -- 0.6 -- -- -- Region III 17.2 68.2 57.2 -- 46.0 -- Region IV 8.3 20.1 36.5 100.0 54.0 100.0
Total 100.0 100.0 100.0 100.0 100.0 t00.0
The optimum solution showed that corn was the only feed grain that would be grown. Production of barley and oats would result in the increased cost of milk production, even in areas not well suited for corn production. In 1963, the production of oats in the Susquehanna River Basin was 19 million bushels. However, the production in oats in Pennsylvania during the period 1951-61 declined approximately 10 per cent. It is probable that a similar trend has existed in the basin over that 10-year period. Barley production during 1963 equaled roughly 2 million bushels and during the past 12-year period, 1951 to 1963, the quantity of barley produced and fed to livestock on producer 's own farms increased by 17 per cent in Pennsylvania. Total barley production in Pennsylvania increased by 43 per cent in the 1951-61 period. However, a large
HOWES: TEST OF A LINEAR PROGRAMMING MODEL ] 3~
proportion of the increase in production was supplied to industry and therefore was not considered in the model.
The optimal solution showed that only alfalfa hay was produced in the basin, whereas, in 1963, the production of clover and t imothy hay equaled 1.3 million tons or 48 per cent of total hay produced. Although the production of clover and t imothy hay in the basin was substantial in 1963, it has been decreasing. In Pennsylvania as a whole, the production of alfalfa increased 2.7 t imes in the period 1951-61, and the ratio of alfalfa hay to clover and t imothy hay increased from .20 in 1951 to .46 in 1961.
The opt imum solution of the model specified production of 3.5 million tons of corn silage to mee t feed requirements for milk production. In 1963, ap- proximate ly 3 million tons of corn silage were produced in tne basin. However , for the state of Pennsylvania, the production of corn silage increased by 35 per cent in the period 1951-61.
The production functions for milk differed substantial ly among the four regions. In all regions of the basin, the production of grains was at the min imum level possible, g iven the constraints of the model. Additionally, the production of corn silage in each region, except Region IV, was at its max imum, given the constraints of the model. In the northern two regions of the basin, relat ively more pasture was used in the production of milk than in the southern two regions.
The production functions for milk in Regions I and I I were similar. In Region I, 4 per cent more of the feed requirement was derived from pasture than in Region II, and, therefore, in Region I, 4 per cent less alfalfa hay was utilized per unit of milk production than in Region II. In Region III, alfalfa hay supplied 33 per cent of the nutr ients for milk production. Relative quantities of feeds required in milk production are shown in Table 3.
The milk production function specified by the model for Region IV indicated an intensive use of its land resources. This pa t tern of production is largely due to its locational advantage in supplying milk to Philadelphia, Wilmington, Baltimore, and Washington, D.C. Ninety-three per cent of the total nutr ients used for milk production in Region IV are outputs of crops which produce
TABLE 3 OPTIMUM SOURCES OF FEEDS FOR MILK PRODUCTION SPECIFIED BY TIlE
LINEAR PROGRAMMING MODEL, SUSQUEHANNA RIVER BASIN
Crop Region I Region II Region III Region IV
Per cent
Corn, grain 30 30 30 30 Alfalfa hay 20 24 33 63 Corn, silage 28 28 28 3 Pasture 22 18 9 4
Total 100 100 100 100
] 34 PAPERS OF THE REGIONAL SCIENCE ASSOCIATION
relatively large quantities of nutr ients per acre. In Region IV, only 3 per cent of the feed requirement for milk was -met by the production of corn silage, and pasture provided only 4 per cent of the total nutrients used in milk production.
Beef production can be subdivided into two distinct types of enterprises, production of feeder cattle and fattening feeders for slaughter. The production of feeders requires that grazing and harvested roughage supply most of the feed nutrients. On the other hand, fat tening cattle for slaughter requires feeding of a ration containing a high proportion of grains. I t is common for the two types of beef enterprises to be located in different areas because of the difference in the types of feed utilized.
The model specified Region I as the opt imum location for the production of feeder cattle. Beef production in Region I would utilize the pasture and harvest forage for which the area is well suited. I t is probable that the relatively small quantities of grain needed would be imported from Region III. However, t ransport costs for grain were not included in the model.
Beef fat tening enterprises under the opt imum solution were located in Region III, where cattle were fed locally produced corn and small quantities of roughage. Again, as in the case for milk, corn was the only grain produced for beef feeding, and total hay requirements for beef production were met f rom the production of alfalfa hay.
The opt imum solution of the linear p rogramming model specified that production of final crops would be concentrated in Region I. In 1963, the production of these crops was quite evenly distributed throughout the basin. The pat tern specified by the model is the result of a comparat ive disadvantage in the production of feed grains in Region I and the locational advantage of Region IV in milk production. The comparat ive advantage in grain production in Regions I I I and IV, given the constraints of the model, resulted in intensive production of feed crops. The alternative of displacing intensive feed crop producing activities in Regions I I I and IV by final crops would result in
TABLE 4 PRODUCTION OF FINAL CROPS IN THE SUSQUEHANNA RIVER
BASIN, BY REGIONS, 1963
Crop Unit Region I Region II Region III Region IV Total
1.000
Wheat Potatoes Sweet corn Beans Tomatoes Cabbage Strawberries Apples
Bu.
Cwt. Cwt. Cwt. Cwt. Cwt. Qt. Bu.
777 5,833
106 66
353 477 819 418
91 442
7 92
12 304 68
5,216 646 153 41
1,007 270
1,926 3,538
3,627 1,340
474 79
873 37
I 2,090 1,216
9,710 8,261
740 278
2,233 796
5,139 5,240
HOWES: TEST OF A LINEAR PROGRAMMING MODEL ] 35
relatively high costs when compared to the displacement of extensive forage producing activities in Region I, because a high positive correlation exists between the production costs of intensive feed crops and production costs of final crops.
The optimal solution of the linear programming model specified that wheat would be produced in all regions of the basiff, with Region I producing 54 per cent of the total. Production of potatoes, according to the opt imum solution, would be limited to Regions I and III, with the latter supplying 78 per cent of the total quantity. The production of peaches, according to the model, would be concentrated in Region III. All other final crops would be produced in Region I. The production of final crops in 1963 and the production pat tern specified by the linear programming model are given in Tables 4 and 5 respectively.
TABLE 5 PRODUCTION OF FINAL CROPS IN THE SUSQUEHANNA RIVER BASIN
SPECIFIED BY LINEAR PROGRAMMING MODEL
Crop
Wheat Potatoes Sweet corn Beans Tomatoes Cabbage Strawberries Apples
Unit
Bu.
Cwt. Cwt. Cwt. Cwt. Cwt. Qt. Bu.
Region I
8,103 1,799
740 278
2,233 796
5,139 5,240
Region II
514
Region III
1.000
869 6,462
i
Region IV
224
Total
9,710 8,261
740 278
2,233 796
5,139 5,240
The pat tern of agricultural land use specified by the linear programming model was substantially different from the pat tern existing in 1963. In general, the model specified a grea ter intensity in the production of grains and hay. On the other hand, the model specified roughly one million more acres of pasture than was used in 1963. The quanti ty of land specified by the model for other crops was not great ly different from 1963 for the basin as a whole.
Differences between the land use pat tern existing in 1963 and that specified by the linear programming model are par t ly explained by the flexibility built into the production functions for milk and beef. Only one grain and one hay crop were specified by the linear programming model, whereas production of three grains and two hay crops was recorded for 1963. Comparisons of land use from the linear programming model and actual land use in 1963 are shown for the basin as a whole and for each production region in Tables 6-10.
A total cost of 135.9 million dollars was specified by the opt imum solution of the linear programming model to produce and transport the required
136 PAPERS OF THE REGIONAL SCIENCE ASSOCIATION
TABLE 6 LAND USE SPECIFIED BY THE LINEAR PROGRAMMING MODEL AND ACTUAL
LAND USE IN THE SUSQUEHANNA RIVER BASIN, 1963, ALL REGIONS, ALL CLASSES OF SOIL
From Linear Land Use Programming Model Actual
Acres
Corn Barley Oats
Total Grains Alfalfa hay
Clover, timothy
Total Hay Corn silage
Pasture
All Feed Crops
485,903
485,903
996,584
996,584 246,258
2,528,832
4,257,577
499,826 116,425 381,289
997,540
592,152
853,076
1,445,228 237,145
1,402,628
4,082,541 Wheat Potatoes Peaches Apples
Total Fruit Snap beans Sweet corn Tomatoes
Cabbage Strawberries
303,641 24,815 7,978
11,004
18,982
5,450 11,759 7,144 2,427
2,056
307,043
35,158 11,280 19,617
30,897 6,491
11,498
111,595 3,240
1,142
Total Other 28,836 33,966
Misc. crops - - 28,417 Idle land 366,343 482,172
Grand Total 5,000,194 5,000,194
agr icu l tu ra l outputs . The cost specified by the model was 18 per cent less than the es t imated actual cost of producing the same ou tpu t in 1963, which was 166.0 mi l l ion dollars. The opt imal solut ion also showed a reduct ion in t ranspor ta t ion costs for mi lk of approx imate ly one mi l l ion dollars, a reduct ion of 9 per cent in the var iable costs of t ranspor ta t ion .
C O N C L U S I O N S
T h e pa t t e rn of agr icu l tu ra l product ion and land use specified by the l inear p r o g r a m m i n g model was grea t ly different from the actual pa t t e rn ex i s t ing in
H O N E S : TEST OF A LINEAR P R O G R A M M I N G MODEL ]37
TABLE 7 LAND USE SPECIFIED BY THE LINEAR PROGRAMMING MODEL AND
ACTUAL LAND USE IN THE SUSQUEHANNA RIVER BASIN, 1963, REGION I, ALL CLASSES OF SOIL
From Linear Land Use Programming Model Actual
Acres
Corn Barley
Oats
Total Grains Alfalfa hay Clover, timothy
Total Hay Corn silage Pasture
All Feed Crops
57,260
57,260 61,800
61,800 56,275
1,530,722
1,706,057
22,071 7,363
149,569
179,003 208,475 461,222
669,697 99,941
948,040
1,896,681 Wheat Potatoes Peaches Apples
Total Fruit Snap beans Sweet corn Tomatoes Cabbage Strawberries
255,554 5,583
11,004
11,004 5,450
11,759 7,144 2,427 2,056
24,020 22,990
464 1,752
2,216 1,661 1,640 1,927 1,952
161
Total Other 28,836 7,341 Misc. crops
Idle land 158,841 212,627
Grand Total 2,165,875 2,165,875
1963. I be l i eve tha t the app l i ca t ion of th is mode l or s im i l a r mode l s to o the r a r e a s wou ld not a p p r o x i m a t e ac tua l p a t t e r n s of p roduc t ion or r e source use. The re fo r e , i t is conc luded tha t the a s s u m p t i o n s upon wh ich the mode l is based a re i nadequa t e for the cons t ruc t ion of mode l s a i m e d a t e s t i m a t i n g the economic f eas ib i l i t y of w a t e r resource i nves tmen t s . F u r t h e r w o r k on the mode l i t se l f is n e c e s s a r y before i t can be p r o p e r l y used in the p red i c t i ve sense r equ i r ed in economic f eas ib i l i t y s tud ies .
In o r d e r to i m p r o v e the model , i t wi l l be neces sa ry to quan t i fy add i t iona l cons t r a in ing re la t ionsh ips tha t a re g e n e r a l l y encoun te red b y the t ype of
] 38 PAPERS OF THE REGIONAL SCIENCE ASSOCIATION
TABLE 8 LAND USE SPECIFIED BY THE LINEAR PROGRAMMING MODEL AND ACTUAL
LAND USE IN THE SUSQUEHANNA RIVER BASIN, 1963, REGION II, ALL CLASSES OF SOIL
Land Use
Corn Barley
Oats
Total Grains Alfalfa hay Clover, timothy
Total Hay
Corn silage
Pasture
All Feed Crops
Wheat Potatoes
Peaches Apples
Total Fruit Snap beans
Sweet corn Tomatoes Cabbage
Strawberries
Total Other
Misc. crops Idle land
Grand Total
From Linear Programming Model Actual
Acres
6,031 3,999 - - 368 - - 13,520
6,031 17,887 7,383 8,564
- - 21,511
7,383 30,075 3,175 4,072
52,312 23,247
68,901
14,455
1 6 , 4 1 6
99,772
75,281 2,508 1,999
181
181 1,949
104
56
54
2,163
17,640
99,772
economic un i t s be ing studied. Care should be t aken not s imply to bui ld empir ica l real i t ies into the model so as to reduce differences be tween the model resul ts and ex is t ing product ion and resource use. Ra ther the type of cons t ra in t s needed appear to require such studies as how economic un i t s t end to react to oppor tuni t ies for increased profits in s i tuat ions where changes in habi ts would be required for the real izat ion of these increased net re turns .
Studies which accura te ly es t imate the economic efficiency of wa te r resource i nves tmen t s need to consider how economic un i t s wil l react to changes in water resources. The use of ma themat i ca l p r o g r a m m i n g models in these
HOWES: TEST OF A LINEAR PROGRAMMING MODEL ] 3 9
TABLE 9 LAND USE SPECIFIED BY THE LINEAR PROGRAMMING MODEL AND ACTUAL
LAND USE IN THE SUSQUEHANNA RIVER BASIN, 1963, REGION III, ALL CLASSES OF SOIL
From Linear Actual Land Use Programming Model
Acres
Corn
Barley
Oats
Total Grains
Alfalfa hay
Clover, t imothy
Total Hay
Corn silage
Pasture
All Feed Crops
285,954
285,954
413,782
413,782
176,139
687,226
1,563,101
237,487
64,984
186,837
489,308
297,631
170,030
467,661
102,134
287,774
1,346,877
Whea t
Potatoes
Peaches
Apples
Total F ru i t
Snap beans
Sweet corn
Tomatoes
Cabbage
Strawberr ies
26,172
19,232
7,978
7,978
165,346
3,255
8,590
12,187
20,777
1,086
2,109
5,186
1,059
535
Total Other - - 9,975
Misc. crops - - 337
Idle land 175,764 245,679
Grand Total 1,792,247 1,792,246
s t u d i e s s e r v e s to m a k e t h i s n e e d m o r e a p p a r e n t .
] 4 0 PAPERS OF THE REGIONAL SCIENCE ASSOCIATION
TABLE 10 LAND USE SPECIFIED BY THE LINEAR PROGRAMMING MODEL AND
ACTUAL LAND USE IN THE SUSQUEHANNA RIVER BASIN, 1963, REGION IV, ALL CLASSES OF SOIL
From Linear Land Use Programming Model Actual
Acres
Corn
Barley
Oats
Total Grains
Alfalfa hay
Clover, t imothy
Total Hay
Corn, silage
Pas ture
All Feed Crops
Whea t
136,658
136,658
513,619
513,619
10,669
258,572
919,518
7,460
236,269
43,710
31,363
311,342
77,482
200,313
277,795
30,998
143,567
763,702
115,169
Potatoes
Peaches
Apples
Total Fru i t
Snap beans
Sweet corn
Tomatoes
Cabbage
Strawberr ies
m
m
6,914
2,226
5,497
7,723
1,795
7,645
4,482
173
392
Total Other - - 14,487
Misc. crops - - 28,079
Idle land 15,322 6,226
Grand Total 942,300 942,300