A test of a linear programming model of agriculture

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 investments 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 mathematical 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).

123

| 24 PAPERS OF THE REGIONAL SCIENCE ASSOCIATION

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 usefulness, 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 assumptions 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 .

] 26 PAPERS OF THE REGIONAL SCIENCE ASSOCIATION

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 production 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.


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 considered 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 equilibrium, 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 production. 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.


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, approximate 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


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


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



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



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


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

A test of a linear programming model of agriculture

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