A mathematical model of the United States beef production system

A mathematical model of the United States beef production system

/IgricuhuralSrstems $ (1980) 295-307 A MATHEMATICAL MODEL OF THE UNITED BEEF PRODUCTION SYSTEM STATES W. C. MILLER Department of Mathematics, Univ...

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/IgricuhuralSrstems $ (1980) 295-307

A MATHEMATICAL MODEL OF THE UNITED BEEF PRODUCTION SYSTEM

STATES

W. C. MILLER

Department of Mathematics, University of Southern Colorado, Pueblo, Colorado 81001, USA & G. M. WARD, T. P. YORKS, D. L. ROSSITER & J. J. COMBS

Department oJ Animal Sciences, Colorado State University, Fort Collins, Colorado 80523, USA

SUMMARY

Dynamic linear programming was used to model the US beef production system, to determine the cattle cycle effects onfeeder cattle supplies and to select feeding options that would maximise USDA choice and prime quality grade beef production. The model represented the US as afire-region bee[production system with inter-regional transportation of off'spring for feeding purposes. Cow weights, offspring weaning weights and/bed efficiency were different for each region, but the feeding activities were the same in each region except/or region two. In region two thefeeding of grain to steer and heifer offspring was not an industry practice. Dynamic properties of the model were achieved through constraints that transferred animals from year zero (initial condition) to year five using calving, cow culling, replacement heifer and death loss rates as controlling parameters. The results of model exercises that used historical parameter values indicate that there will be a significant decrease in the supply of beef at all grade levels over the next five years. They also suggest that it is more economically efficient to feed calves from the southeastern and northern parts of the USA if they are transported to the southwestern region.

INTRODUCTION

Beef production represents the largest single agricultural enterprise in the United States and beef accounts for by far the largest single item in the food budget of consumers--an average of 2.5% of disposable income or about 15% of food expenditures. The beef production system is fragmented more than any other food 295 AgriculturaISystems 0308-521X/80/0005-0295/$02.25 © Applied Science Publishers Ltd, England, 1980 Printed in Great Britain

296

W. C. MILLER, G. M. WARD, T. P. YORKS, D. L. ROSSITER, J. J. COMBS

production system, with calf production dependent generally upon low cost grazing resources and the finishing phase upon the use of feed grains and/or silage. The growing phase between weaning and finishing varies with the region of the country, the demand for beef and the price of feed grains. The supply of beef is not closely related to cattle numbers in a given year but rather depends upon, in addition to a variety of economic factors, the phase in the cow numbers curve and the beef production curve. Wide fluctuations in cattle numbers in the United States have been known to exist since at least the late 1800s (Burmeister, 1949; Breimyer, 1955). The fluctuations were thought to be due to herd build-up encouraged by economic and/or feed conditions. Economic analyses of the fluctuations have been carried out by many economists, including Enrich (1966), Knox (1975) and Gee (1978). In more recent times, periods between successive peaks or troughs for beef cattle numbers have been a somewhat uniform ten years. This study was initiated to develop a mathematical model of the beef cattle industry that satisfies the following objectives: (1) (2)

(3)

To provide a model which would represent the dynamic nature of the US beef cattle industry from cow conception to carcass beef. To provide a model for determining weaned calf supplies over a five-year period as functions of initial condition cow herd size and annual calving, cow culling and replacement heifer rates. To provide a model for determining feeding options that will maximise USDA choice grade beef production, the grade that provides the optimum financial return to the producer from annual supplies of weaned calves within the practices and constraints of the industry.

MATERIALS AND METHODS

For the purposes of the model the United States was divided into five beef-producing regions. The northeastern United States, where only 1 ~o of the beef cows are located, was not included. The regions were determined by analysis of environmental factors affecting beef production (Nix, 1976). Figure 1 shows the regional boundaries as used by Nix (1976) and Table 1 (year zero) the number of beef cows that had calved in each region on 1 July, 1976. These beef cow numbers provided the initial condition herd size in year zero. Calving, cow culling and replacement heifer rates in Table 2 were the model parameters for determining herd size in subsequent years. These parameter values were determined from the data of Nix (1976) and data reported by the American National Cattlemen's Association (1975). Brood cow weights and offspring weaning weights used in the model were different for each region (Nix, 1976) and thus the feed requirements also. Feed efficiencies and

"~ "~

Nevada Utah

"~

Fig. I.

,,,zooo

Colifornio

~.

\

,da~o

/

(

Woshinqton Oreaon

-

~

Oklahoma

New M e x i c o

I

4

,~OUTH PLAINS

REGION

North Dokoto South Dakota Nebraska

Co,o,odo

Montana Wyoming

J

I

/ I

-"-7

l

/ /

ALABAMA ARKANSAS GEORGIA MISSISSIPPI TENNESSEE LOUISIANA

Minnesota Wisconsin Michigan Illinois "~Indiono

REGION I CORN BELT

2

NORTHCAROLINA SOUTHCAROLINA FLORIDA

VIRGINIA KENTUCKY

REGION

SOUTH E A S T WEST VIRGINIA

Ohio Iowa

Regional breakdown of United States as used in the beef production analysis.

IN ' ; ; R M O U N T A I N " ~

)

f

REGION 3 T____.HH N0 R PLAIN..__S

o0

Z

C ,-t

t~ ,.[1

t'-

0 m

t'-

>

~r > -] tsl >

298

W. C. MILLER, G. M. WARD, T. P. YORKS, D. L. ROSSITER, J. J. COMBS TABLE 1 BROOD COW NUMBERS × l0 6 BY REGION AND YEAR

Year 0 1 2 3 4 5

1

2

Region 3

4

5

7.604 7"883 8.137 9-035 10.588 12"838

11.138 10.995 11.313 12.450 14.475 17-701

10'019 9.985 10-256 11"281 13.092 15.975

9-809 9"782 10-051 11"069 12.879 15.767

3.677 3-595 3.668 4"010 4.641 5.563

weight gains were in terms of total digestible nutrients (TDN) determined from work by Fox & Black (1977). Feed requirements were entered in the model as technological coefficients. Details of these calculations have been reported by Combs (1978). Six ration options for both steers and heifers and two options for feeding bulls were provided for each region and any combination of rations could be chosen for any of the five years. Heifers were slaughtered at 80 ~ of the weight of steers, which corresponds to the same carcass composition for each sex (Fox & Black, 1977). The model also accounts for the beef contribution from culled cows and bulls. Ration options, which included adequate supplements of protein, minerals and vitamins, were grazing with and without protein supplement and grazing followed by finishing with and without supplement, corn silage and corn silage followed by finishing on concentrate diets. These management options are shown diagrammatically in Fig. 2. Concentrate finishing options were not included for the southeastern region because TABLE 2 CALVING, CULLING AND REPLACEMENT RATES BY REGIONAND YEAR

Parameter Year

1

2

Region 3

4

5

Calving

0 1 2 3 4

0.82 0.84 0.86 0.88 0.89

0.75 0.77 0-79 0.81 0-82

0.77 0.79 0.81 0-82 0.83

0.77 0.79 0.81 0.82 0.83

0.77 0.79 0.81 0.82 0.83

Culling

0 1 2 3 4

0.11 0.08 0.05 0.02 0.01

0.13 0.10 0-07 0.04 0-01

0.15 0.12 0.09 0.06 0-03

0.12 0.09 0.06 0.03 0.02

0.11 0.08 0.05 0.02 0-01

Replacement

0 1 2 3 4

0.14 0.19 0.24 0.29 0.31

0.15 0-20 0.25 0.30 0.32

0-17 0.22 0.27 0.32 0.34

0.14 0.19 0.24 0.29 0.31

0.12 0.17 0.22 0.27 0.29

MATHEMATICAL MODEL OF US BEEF PRODUCTION SYSTEM

299

COW

r

Weoned Calves

Ration Options-

1

2

I

Silage to 335 kg

r

Conc.

Slaughter

Fig. 2,

3

5

6

Grazing

I

Grozing + Suppl to 335 kg

Grazing only

Conc.

Conc.

Gr~ing 4Suppl.

I

Silage

4

to

33~ kg

Steers

500

500

500

500

400

400

Weights ( kg ) Heifers

400

400

400

400

320

320

Feeding options available in the model and the resulting slaughter weights for each.

the practice is not common there. The model also included inter-regional transportation options for offspring entering the finishing phase. Beef carcass quality was divided into three categories: (l) USDA choice and prime, (2) USDA Good and Standard and (3) other or commercial grade. The grade was determined by ration options and age of cattle.

300

W.C.

MILLER, G. M. W A R D , T. P. YORKS, D. L. ROSSITER, J. J. COMBS MODEL STRUCTURE

The model structure can be described by the following inputs and results.

Model inputs (1) Number of brood cows, sires and replacement heifers in each of the five (2) (3) (4) (5) (6)

production regions for year zero. Calving, culling and replacement rates for the cow herd for each of the five years. (Production parameters.) Total digestible nutrient requirements for all classes of animal in the model. Weaning weight of steers and heifers in each region. Transportation costs for weaner calves from any region to any other region. Quantity and quality grade of beef that each class of animal would produce.

Model results (1) Determine the cow, replacement heifer, sire and offspring herd size for each

(2)

region for each of the five years. Determine the transportation and feeding activities for each region for each of the five years that would maximise first quality grade beef production. MATHEMATICAL FORMULATION

The variables were denoted by Bi~k, FMok, FFm~k, Tnp~k,P,~k, D~k and Mqj. Where Bigk denoted the breeding activities. i i i j

= = = =

1 for brood cow 2 for sire 3 for replacement heifer 0, 1. . . . . 5 for the year in which the activity occurred and k = 1,2 . . . . . 5 for the region of the activity.

FMok denoted the male feeding activities. l = 1 for bull calf fed corn silage and hay from weaning to 300 kg and finished on grain to 495 kg. l = 2 for bull calf fed corn silage and hay from weaning to 300 kg and finished on grain to 585 kg. 1 = 3 for steer fed corn silage and hay from weaning to 300 kg and finished on grain to 495 kg. 1 = 4 for steer fed on silage to 495 kg. l = 5 for steer fed on grass to 395 kg. l = 6 for steer fed on grass and supplement to 395 kg. l = 7 for steer fed on grass and finished on grain to 495 kg.

MATHEMATICAL MODEL OF US BEEF PRODUCTION SYSTEM

301

l = 8 for steer fed on grass with supplement and finished on grain to 495 kg.

FFm~kdenoted m m m m m m

= = = = = =

1 2 3 4 5 6

for for for for for for

the female feeding activities.

heifer heifer heifer heifer heifer heifer

b a c k g r o u n d and finished on grain to 395 kg. fed silage to 395 kg. fed grass to 315 kg. fed grass and supplement to 315 kg. fed grass and finished on grain to 395 kg. fed grass with supplement and finished on grain to 395 kg.

TNpik

denoted the transportation activities that transported male and female offspring inter-regionally for feeding activities. n - - 1 for males n = 2 for females and p = 1, 2 . . . . .

5 the region o f origin, p # k.

P,~kdenoted the n u m b e r o f tons o f corn feed equivalent units* o f feed purchased in year j for region k. r r r r r

= = = = =

1 2 3 4 5

for for for for for

corn silage grass hay supplement

Mqj denoted the n u m b e r o f tons o f quality grade meat produced in year j. q = 1 for highest quality (choice and prime) q = 2 for second quality (good and standard) q = 3 for lowest quality (commercial)

Djk denoted

the n u m b e r o f dollars purchased in year j for region k.

The d y n a m i c linear p r o g r a m m i n g m o d e t o o k on the following form: 2

Maximise:

5

8

5

6

5

z=~b~jkBijk+~Zb~jkFMtjk+~b~jkFF, i=1

k=l

/=1

k=l

m=l

k=l

njk j=1,2,...,5

Subject to the following constraints. * Corn feed equivalent unit (CFU) is a term developed by the US Department of Agriculture for the expression of feeds on a common basis. One CFU is equal to 1 lb of No. 2 corn containing 78-2%TDN (total digestible nutrients).

302

w.c.

MILLER, G. M. WARD, T. P. YORKS, D. L. ROSSITER, J. J. COMBS

Capital constraint set: 3

5

i=1

5

8

j=O k=l 6

m=l

5

5

k=l

I=l

j=l

5

5

j=l

k=l

2

n=l

5

p=l

5

5

j=O k=l

5

5

5

3

5

r=l

j=l

k=l

q=l

j=l

j=l,2 k=l,2

..... .....

5 5

d Where the superscripts a, b, c, d, e a n d f o f the C~k, C~,~k, C.pjk, C~ik and C~ cost coefficients of the variables allow for various cost figures.

(1)

Year zero brood cow, sires and replacement heifer constraint set:

Biok = Nik i=1,2,3 k=l,2 .....

5

where Nik denoted the number of brood cows, sires and replacement heifers in region k for year zero. (2)

Feed constraint set: 3

8

6

~a~jkBijk+~a~jkFMuk+Za3jkFFmjk i=1

2

+

m=l

I=1

5

p=l

r=l

r=l,2 j=l,2 k=l,2 (3)

..... ..... .....

5 5 5

Beef production constraint set: 2

5

8

5

6

5

bq3jkFF,,,jk + Mq~ = 0 i=1

k=l

/=1

k=l

m=l

k=l

q=2,3 j=l,2 .....

5

where bq~jk, ~ b~tjk 2 and bq,.~ 3 k were the tons of each of the quality grades of carcass beef produced by all of the k regions in each year j.

303

M A T H E M A T I C A L M O D E L O F US BEEF P R O D U C T I O N SYSTEM

(4)

Constraint set that transfers male offspring from the brood cow activities in the year j - 1 to the feeding activities in year j for each region, k 8

I=l

5

p'=l

p'¢k

5

k'=l

k'¢p

p=l,2 j=l,2 k=l,2

..... ..... .....

5 5 5

wheref~_ ~) was the fraction of the kth region's female calf crop to enter the feeding program. (5)

Constraint set that transfers female offspring from the brood cows' activities in year j - l to the feeding activities in year j for each region, k: 6

ra=l

5

p'=l

p'¢k

5

k'=l

k'~p

p=l,2 j=l,2 k=l,2

..... ..... .....

5 5 5

wheref~J_ ~ was the fraction of the kth region's female calf crop to enter the feeding program. (6)

Constraint set that transfers brood cows and replacement heifers from year j - 1 brood cow and replacement heifer activities to year j brood cow activity for each region, k __ f(2_ l)kBltj_ l)k -- 0"98B3jk q- Bljk = 0

j=l,2 k=l,2

..... .....

5 5

W here

2 X)k was the fraction of the kth region year j - 1 brood cow herd after f~_ culling and death loss.

(7)

Constraint set that transfers replacement heifer offspring from the brood cow activity in y e a r j - 1 to the replacement heifer activity in y e a r j for each region, k. __f(3_ 1)kBlj k + B3jk = 0

j=l,2 k=l,2

..... .....

5 5

where f j-l)k 3 was the fraction of the kth region's female calf crop saved for replacement heifers.

304

W . C. MILLER, G. M. W A R D , T. P. YORKS, D. L. ROSSITER, J. J. COMBS

(8)

Constraint set that transfers sires and replacement sires from yearj - I to year j for each region, k: - m 6 _ 1 ) k n l l j _ l)k - - 0 ' 6 7 B 2 1 j - Ilk + B2jk = 0 j=l,2

.....

5

k=l,2

.....

5

where m~_~tk was the fraction of the kth region's male calf crop saved for replacement sires. MODEL OUTPUT

The model output consisted of the simulated brood cow, replacement heifer and feeder cattle herd sizes for years one to five and these are listed by region and year in TABLE 3 REPLACEMENT HEIFER N U M B E R S

Year

× 10 6 BY REGION A N D YEAR

Region

0 1 2 3 4 5

I

2

3

4

5

1"559 1"065 1"498 I '953 2-620 3-282

1-559 1-671 2-199 2-828 3.735 4"632

1-703 1"703 2"197 2"769 3"610 4"451

1-373 1'373 1'858 2-412 3.210 3'993

0"404 0-441 0-61 I 0"807 1'083 1-346

TABLE 4 FEEDING O P T I O N S SELECTED A N D N U M B E R S OF H E A D

Region

Sex

Year 1

M F M F M F M F M

F

× l 0 6 BY REGION, YEAR A N D SEX

None None None None None None Option 6.660 Option 4.456 Option 5.095 Option 4.021 Option 3.128 Option 2-506

7 6 3

2

3

None None None None None None Option 8 6.838 Option 5 3.798 Option 7 9-139

None None None None None None Option 7.191 Option 3.191 Option 5-330 Option 4.254 Option 2-040 Option i.640

7 1 6

Option 1 2.522 Option 6 1.034

4

7 5 3

5

None None None None None None Option 7 8-054 Option 5 2.672 Option 7 10.676

None None None None None None Option 7 9-530 Option 5 2.770 Option 7 12.688

Option 1 2.867

Option 5 2.848

7 1 5

305

MATHEMATICAL MODEL OF US BEEF PRODUCTION SYSTEM

TABLE 5 BEEF PRODUCTION IN METRIC TONS × 10 6 BY REGION AND GRADE

USDA Grade

Choice and prime Good Commercial

1

2

Year 3

4

5

4-378 2.952 1.003

4.194 2.799 0.814

4.130 2.743 0.599

4.265 2.845 0.435

4.913 3.277 0.462

Tables 1 and 3. The feeder cattle numbers then became the right-hand side values for the supply of steers and heifers in the feeding activities section. Feeding options were selected to maximise the production of USDA choice or prime quality beef. The feeding options selected and the regions wherein the cattle were fed by year are listed in Table 4. Table 5 lists the resultant beef production by grade and year.

DISCUSSION

The calving, culling and replacement rates listed in Table 2 were determined from studies by Nix (1976), who investigated producer practices by region, and from the American National Cattlemen's Association (1975) which reported producer practices during recent cattle cycles. The parameters listed in this Table are subject to speculation because it is never precisely known what practices producers will follow in herd culling and replacement heifer retention or what specific effects herd care, disease, prices and weather will have on calving rates. Nonetheless, historical production parameters can be an indicator of future production parameters. Reported practices, as well as the fact that the brood cow herd liquidation from the 1975 peak has been more severe than in past cycles, were the factors considered in developing the numbers in Table 2. Tables 1 and 3 show that, beginning in the first year and continuing to the third year, modest increases in brood cow and replacement heifer herd sizes are projected. By the fifth year brood cow and replacement heifers are still increasing, indicating a greatly expanded beef production potential in subsequent years. The relative distribution of cows and heifers by geographical region was not changed because the assumption was made that feed supplies were adequate to support any increases in the size of the breeding herd. In reality, future expansion of beef cattle numbers will be influenced bydifferential marginal costs to expand forage supplies. The model allowed inter-regional movement of weaned calves in response to more economical finishing to simulate actual practice within the United States. The requirement to maximise production of the choice and prime grades of beef was introduced because the financial return to producers is normally considerably better than for lower grades. The requirement for the model to maximise production of

306

W. C. MILLER, G. M. WARD, T. P. YORKS, D. L. ROSSITER,J. J. COMBS

these grades resulted in the selection of high concentrates or, secondly, silage rations for finishing cattle. Grazing systems without concentrates will not produce carcasses of the higher grades. Feeding systems selected represent the more common practices in the USA except during periods of large cattle inventories when so-called 'grassfed' beef appears on the market. Bull feeding as an option was also excluded because, although beef production per cow unit can be increased, the product will not meet grade requirements. Regional allocation of calves to feed in the southern plains and inter-mountain regions was dictated by the more favourable returns to feeding in these regions as described by Old & Combs (1978) and by Fox & Black (1977). Almost no finishing of cattle occurs in the southeastern regions but cattle feeding is an important enterprise in the Corn Belt and northern plains states. The introduction of suitable parameters to the model could, of course, produce distribution of cattle finishing more in line with actual practice in the United States. Table 5 shows that there is a significant decrease in the production for all grades of beef until year four when production begins to increase due to decisions in years zero or one (see Table 3) to increase the replacement heifer herd and the brood cow herd. The decrease in beef production during years one, two and three is the result of more female ol,'spring going to the replacement herd, rather than to the feedlot, and the reduction of high culling rates for the brood cow herd that was characteristic of the herd liquidation phase of the previous cattle cycle. The model can be used to examine any combination of assumptions similar to those described. For example, constraints upon certain feeds such as grains could alter beef quantity and quality over a period of years. Changes in geographical advantages would shift production between regions. The model has been used to examine the impact of three heifer replacement rates upon the ten-year projection of cattle numbers, feed requirements and beef production (Miller et al., 1979).

REFERENCES AMERICANNATIONALCATTLEMEN'SASSOCIATION(1975). Cattle Economics Report. (August). AMERICANNATIONALCATTLEMEN'SASSOCIATION(1977). Cattle Economics Report. (February). BREIMYER,H. F. (1955). Observations on the cattle cycle. Agricultural Economics Research, 7, 1. BURMEISTER,C. A. (1949). Cycles in cattle numbers. Livestock and Meat Situation, 28, 9. COMICS,J. J. (1978). Systems Approach to Forage Fed Beef. MS Thesis. Colorado State University, Fort Collins. ENRICH,R. L. (1966). Econometric Analysis of the UnitedStates Beef Cattle Cycle. ScienceMonograph 1. Wyoming Agricultural Experiment Station. April.

Fox, D. B. & BLACK,J. R. (1977). A System for Predicting Performance of Growing and Finishing Beef Cattle. MSU Research Report No. 328. GEE, K. (1978). Factsfor Colorado Feeder Cattle Producers. Economic Department, Colorado State University, Fort Collins. KNox, P. L. (1975). The United States Beef Cattle Industry: A Policy Analysis with Cost of Adjustment Theory. PhD Thesis, Colorado State University, Fort Collins.

MATHEMATICAL MODEL OF US BEEF PRODUCTION SYSTEM

307

MILLER, W. C., ROSSITER,D. L., WARD,G. M. & YORKS,T. P. (1979). Projections of the cattle cycle in the 80's. J. Amin. Sci., 49(3), 629. Nix, J. E. (1976). Estimated production and expenses for beef cow-calf enterprises in five regions of the US. Livestock Meat Situation. USDA, 210, 39. OLD, C. A. & COMBS,J. J. (1978). Regional advantages for cattle feeding. Feedstuffs, 50, 13.