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A time-dependent bioeconomic model of commercial beef breed choices
Agricultural Systems 45 (1994) 331-347 © 1994 Elsevier Science Limited Printed in Great Britain. All rights reserved 0308-521X/94/$7.00
i!i,I ELSEVIER
A Time-Dependent Bioeconomic Model of Commercial Beef Breed Choices* Bryan E. Melton Department of Economics and Animal Science, 467 Heady Hall, Iowa State University, Ames, Iowa 50011, USA
W. Arden Colette Department of Agricultural Business and Economics, West Texas State University, Canyon, Texas 79016, USA
Kenneth J. Smith Department of Animal Science, Texas Tech University, Lubbock, Texas 00000, USA
&
Richard L. Willham Department of Animal Science, Iowa State University, Ames, Iowa 50011, USA
(Received 10 May 1993; revised version received 19 July 1993; accepted 20 July 1993)
A BSTRA CT The breed choice decision requires simultaneous consideration of both economic and physical relationships in light of the multi-year, multiproduct nature of commercial cow-calf production. To accomplish this a time-dependent bioeconomie model of cow-calf production was developed in which both physical output-input relationships and production decisions are expressed in terms of cow age and stage of production. The breed choice and optimal herd distribution (culling age and proportion by * Journal series paper #J-15452 of the Iowa Agricultural and Home Economics Experiment Station, Project #3175. 331
332
B. E. Melton, W. A. Colette, K. J. Smith, R. L. Willham age) decisions are then cast within the economic framework o f maximizing net returns per unit of nutrient resource derived from a fixed land area. The resulting bioeconomic model is easily solved, as illustrated by an empirical example evaluating 16 alternative breed choices for a representative West Texas ranch. Results indicate a marked economic preference for smaller, higher productivity animals, such as Sahiwal. Conversely, large, slow maturing breeds with marginal reproductive capacity, such as Charolais, are least profitable for the fixed resource area defined.
INTRODUCTION In the last 15 years the number of beef breeds available to commercial producers in the US has increased many fold. Producers are, therefore no longer restricted to Hereford, Angus, or animals of similar genetic composition in their breed choice decisions. They may now select from an array of breeds representing an unprecedented range of biological types and genetic abilities. While such diversity enhances the flexibility of commercial producers to adapt to changing market and economic conditions, it also complicates the breed choice decision. Commercial producers, marketing primarily weaned feeder calves, must now make breed choice decisions that balance the current and long-term productivity of a potentially diverse cow herd against limited production resources, such as land area, in light of prevailing and expected economic and market conditions. Such decisions require simultaneous consideration of multiple facets of the cow-calf production process. The most notable of these, in addition to resources available (especially with respect to forage nutrient production), market prices, and prevailing economic conditions, are the following: (1) the multi-product, multi-year nature of beef cows in which each cow in the herd is expected to produce a weaned calf in each year of her economic life as well as becoming an additional marketable product upon culling; (2) differences between breeds in annual reproductive (calving rate and weight) and productive (weaning rate and weight) performance; and (3) differences between breeds in nutrient requirements for production and reproduction including the cow's own weight maintenance and growth requirements. In this paper a time-dependent breed evaluation model is developed relecting both physical and economic considerations in commercial breed choice. Applications of this model to the breed choice decision are then illustrated for conditions representative of commercial cow-calf production in the range areas of the West Texas Panhandle.
Time-dependent bioeconomic model of beef breeds
333
M O D E L SPECIFICATION Cow herd nutrient requirements, by stage of production, are central to breed choice decisions. Larger animals tend to produce larger progeny, but have simultaneously greater nutrient requirements for maintenance, growth, and production. Hence, for a fixed land area, implying fixed levels of forage nutrients available, producers must balance animal size and per head nutrient requirements against animal numbers and herd size. The variables required to reflect these factors in the model include various measures of animal performance, weight, and nutrient requirements, as well as variables reflecting the prevailing economic and resource conditions. In defining the variables for weight, weight gain, and nutrient requirements, the convention of using upper case letters to refer to the parent (cow) or herd and lower case to refer to the progeny (calf) is adopted. Furthermore, an additional variable, y, is added to calf variable names whenever necessary to reflect the age, in years, of the calf's dam. Additional variable naming conventions include the use of an over-bar (-) to represent aggregate herd averages of the variable across animal ages (in years), a hat (^) to represent adjusted values of the variable in question, dot ( ' ) notation replacing the breed component of the variable name to denote averages across breeds, and substitution of years (y) for days (t) in the variable name to indicate the summation (integration) of daily values to an annual basis. In light of the large number of variables and variations involved in the development of the model, only root variable definitions for the individual cow are summarized in Table 1.
Animal nutrient requirements Energy and protein are the primary nutrients of concern under range grazing conditions. A cow's daily nutrient requirement for M E may be written as the sum of her requirements for each biological function performed: maintenance (m), growth (g), lactation (/), and reproduction (p). Thus ME(i, t) = MEre(i, t) + MEg(i, t) + MEl(i, t) + MEp(i, t) (1)
Cow daily nutrient requirements Various estimates of these daily nutrient requirements, by biological function, exist in the literature of animal science.. Among these are Neville & McCullough (1968), who have estimated the daily M E requirement for maintenance and growth (weight gain) of post-pubescent cows in temperature neutral environments to be MEre(i, t) + MEg(i, t) = 0"1374W(i, 00.75 + 6"3G(i, t)
(2)
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B. E. Melton, W. A. Colette, K. J. Smith, R. L. Willham
TABLE 1 Variable definition
Variable
Units
Definition
ME(i, t)
mcal/day
ME,..g,t.p(i, t)
mcal/day
DP(i, t)
kg/day
DPm.g,l,p(i, t)
kg/day
W(i, t) G(i, t) Ai Bi
kg kg/day kg kg
Total daily metabolizable energy (ME) required by an animal of the ith breed on day of age, t. Daily M E required for maintenance (m), growth and weight gain (g), lactation (l), or reproduction and pregnancy (p), respectively, by an animal of the ith breed on day of age, t. Total daily digestible protein (DP) required by an animal of the ith breed on day of age, t. Daily DP required for maintenance (m), growth and weight gain (g), lactation (/), or reproduction and pregnancy (p), respectively, by an animal of the ith breed on day of age, t. Weight of an animal of the ith breed on the tth day of age. Weight gain of an animal of the ith breed on the tth day of age. Asymptotic mature weight of an animal of the ith breed. Asymptotic total weight gain of an animal of the ith breed. Growth curve rate of growth parameter of an animal of the ith breed. Weight of milk produced by a cow of ith breed on the tth day of lactation. Parameter of daily lactation of an animal of the ith breed. Parameter of the rate of change in daily lactation of an animal of the ith bread. Prenatal weight of an embryo of the ith breed on the tth day of gestation (p). The prenatal rate of growth for an animal of the ith breed. Percentage death loss of an animal of the ith breed that is y years of age as of calving. Percentage weaning rate of an animal of the ith breed that is y years of age as of calving. Age adjustment factor (proportion of maturity) for an animal of the ith breed that is y years of age as of calving. Number of cows of the ith breed and yth age in years, as of calving, in a herd. The probability distribution function of animal age (proportion of animals in herd by years of age) for the ith breed. Optimal culling age, in years, for the ith breed. The net present value of the nth generation of an animal of the ith breed when each successive generation is acquired at b years of age and culled at s i years. The net annual returns of an animal of the ith breed at y years of age, as of calving. The market value of an animal of the ith breed at y years of age, as of calving. The applicable discount rate. The applicable rate of (exogenous or endogenous) genetic progress due to selection. Average annual returns per unit land area for the ith breed when optimally culled and stocked. Total ME provided per unit land area. Total DP provided per unit land area. The stocking rate of the ith breed when optimally culled and land resources are fully utilized.
Ki
M(i ,tt)
kg/day
Ci
kg
li w(i, tp)
kg
0i DL(i, y)
%
WR(i, y)
%
AA(i, y) N(i, y)
~i(Y) si C(i, b, si, n)
Y $
NRi(Y)
$
M(y)
$
P Y
R(s3
$
ME*
mcal/ha kg/ha hd/ha
DP* ~ri
Time-dependent bioeconomic model o f beef breeds
335
The additional daily M E required for lactation has also been estimated (Neville & McCullough, 1968) as
MEl(i, t)
~0.41W(i, t)°75+ 1.9G(i, t)+O.7433M(i, tt) 0
/
(3)
0 tt > tt* where the total lactation cycle is tt* days in length. ,Reproductive M E requirements for prenatal growth are similarly estimated (Moe & Tyrrell, 1971) as
Ot* where the total gestation period is t* days, (W(i, t I tp = 0) = cow weight on the day of conception (tp -- 0), and exp ---- the base of the natural logarithm function. Substituting eqns (2) to (4) into eqn (1) yields a total daily M E requirement function expressed in terms of only three breed-specific variables: W(L t), G(L t), and M(L it) plus time (days of age, lactation, or gestation). Further simplification of the daily M E requirement function is made possible by observing that normal weight and weight gain on any day of age may be represented by a growth function of the form (Brody, 1945) MEp(i, t)
j0.000 567 exp(O.O174tp)W(i, t I tp=0) °75 / 0
W(i, t) + A i - B i exp ( - - K i t )
(5)
where
d W(i, t) G(i, t) - - dt
-
KiB i
exp ( - Ki t)
(6)
Furthermore, M(L tt) may also be expressed in terms of a lactation cycle function (Jenkins et al., 1991)
M(i, tl) -
Ii
6",.exp (lttl)
(7)
Substituting eqns (5) to (7) into eqn (1) allows the total daily M E function for post-pubescent cows to be reduced to a function of only time: day of age (t), day of lactation (it), and day of gestation (tp). Differences in the estimated daily M E requirements between cows, or breeds of cows, of comparable age and production status are then represented directly by differences in a limited number of breed-dependent parameters (At, Bt, Ki, C t and 1~) associated with estimated growth curves and lactation cycle functions (eqns (5) and (7)). Maintenance protein losses are dominated by fecal loss (approximately 80%) according to N R C (1984) recommendations. Given a fecal protein loss of 3.34% of dry matter intake and assuming a maximum average
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B. E. Melton, W. A. Colette, K. J. Smith, R. L. Willham
forage dry matter intake of 2% of body weight, the DP requirement for maintenance is approximately 0.000 84 times body weight. DP requirements for weight gain and milk production are taken directly from NRC (1984, p. 9) where the weight gain requirement function reported by NRC is one adapted from Fox. However, NRC (1984) recommendations for 55 g of DP per day in the last trimester of pregnancy are re-specified to a continuous function of average fetal weight over the last trimester, resulting in an average requirement of 0-0023 kg per kg of fetal weight and an average third trimester fetal weight of approximately 24 kg. Estimates of total daily protein requirements can, therefore, be derived in terms similar to those used in the ME functions as follows: DPm(i, t) + DPg(i, t) = 0"000 84 W(i, t) + G(i, t) (8) [0"235 - 0"000 26W(i, t)]
DPt(i, t) = { 0.0335M(i,0 tl)
0 < t/ - tt*
(9)
t l < tff
DPp(i, t) -- {0-0023w(i,0 tp)
O
(10)
Assuming that progeny weight on any day of gestation can be expressed as an exponential function as weight increases from zero at conception to birth weight (A i - B), w(i, tp) = exp (O~tp) (11) where 0i = In (At - B~)/t* also allows the DP(i, t) functions to be expressed in terms of time and the parameters of growth and lactation.
Calf daily nutrient requirements While progeny may receive the majority of their nutrient requirements from the cow while nursing, over the course of a typical lactation cycle the calf will frequently acquire additional nutrients. Similarly, calves to be retained for replacement in the breeding herd have nutrient requirements from weaning to puberty which must also be fulfilled. NRC (1984) recommendations suggest that eqn (8) is adequate for calf protein requirements, but that a higher energy level is required than would be provided by eqn (2). Accordingly, calf daily ME requirements are estimated on a female basis from NRC (1984) recommendations as
me(i, t) = -29.3039 + 0.3888w(i, t) °7s + 41-7419g(i, t) (0.004 34) (0.002 21)
-10-8340g(i, 0-094 84
t) 2
(0.510 30) r 2 = 0.9999
(12)
Time-dependent bioeeonomicmodel of beef breeds
337
for t < 600, an arbitrarily selected age which should exceed the onset of puberty (standard errors are shown in parentheses below the estimated regression coefficients). Because males tend to grow faster and to a larger size than females, additional nutrient adjustments are required for sex of progeny. N R C (1984) values suggest that this adjustment should be plus 12% for preweaning (intact) male calves. (For simplicity, no males are assumed to be retained after weaning as replacement sires are acquired exogenously in most commercial herds.) While nursing, the progeny's requirements for both ME and DP must then be reduced by the nutrient yield of the dam's milk, obtained by multiplying eqn (7) by 0.564 and 0.030, respectively (NRC, 1984); subject to the conditions that the result is non-negative.
Adjustments to cow--calf-daily nutrient requirement functions Additional adjustments to the daily nutrient requirements are needed to reflect that not all cows will wean a calf each year; that some cows will die; and that performance, with respect to both these characteristics as well as production parameters, such as calf weight and milk production, will be influenced by cow age. Annual death losses and weaning rates, in each year of an average cow's life (y), without respect to breed, are estimated from data developed by Rogers (1972) and Bentley et al. (1976), respectively, for cows 2 to 15 years of age as follows: /-2.4078 +0.32269y~ DL(.,y < 10)xl00 -- 2 + exp 1` (0-2155) (0.023 73)]
r2=0.9585
(13a)
[ 6 . 4 0 0 2 - 0.55421y] DL(.,y< 10)xl00 = 7 - exp 1,(0.161 34) (0.02373)]
r2=0.9672
(13b)
WR(.,y< 10)xl00= 74.9647 + 6.6532y - 0.586 01y2 r2=0.9956 (0.74006) (0.31010) (0.02206)
(14)
and
Breed differences for weaning and death loss rates, by age of dam, are generally not reported. Hence, the adjustment required to obtain these breed-specific values over time must be based on point in time estimates relative to the above equations. For example, weaning rates for alternative breeds may be reported on an age constant basis (typically seven years of age). Equation (14) may then be used to estimate breed-specific weaning rate differences at any age based upon the ratio of reported point in time (y*) weaning rates for the ith breed (WR(i, y*)) to the predicted weaning rate from eqn (14) at the same age (WR(., y*)), i.e.
wR(i,
WR(i, y) = ~ . ,
'ly*)
~ WR(., y) y'))
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B. E. Melton, IV. A. Colette, K. J. Smith, R. L. Willham
Multiplicative age of dam adjustments, as a percent of mature production, are estimated using monthly additive adjustments reported by Wilcox et al. (1971)
AA(., y) = 0.4959 + 0.1649y - 0.0164y 2 + 0.00046y 3 r 2 = 0.9909 (0-00796) (0.00834) (0.001 10) (0.00004)
(15)
These relationships are used to adjust both M E and DP requirements. For purposes of illustration, the expected daily M E requirements for a cow of age y, adjusted for age, weaning rate, and death loss assuming a uniform distribution, are shown below:
M^E(i, t) = {1 - [t - 365y] DL(i, Y~)}365 x { MEre(i, t) + MEg(i, i) + WR(i, y) × [MEt(i, t) + MEp(i, t)]}
(16)
where
MEt(i,t)= ~O.O41W(i, t)°75 + l.9G(i, t)+ AA(i,y)O.7443M(i, tl)
O
0
tl > t~
and
MEp(i, t)= ~AA(i, y)0.000567 exp(O.O174tp)W(i, t [ tp = 0)0.75 0 < tp
o
tp>¢
The corresponding net (of milk production) daily pre-weaning M E requirements on the ith day of age for the progeny of a cow y years of age are similarly adjusted to obtain expected daily values as follows:
me(i, t, y) - AA(i, y) 0.569M(i, t) = -29.3039 + 0.3888[AA(i, y)w(i, t)] °75 + 41-7419g(i, t) (0.004 34) (0.002 21) (0.150 30) -10.8340g(i, t) 2 - AA(i, y)O.569M(i, tl) (0.09484)
(17)
which are expressed on a per cow basis when multiplied by WR(L y). Comparable expectations may then be computed for DP.
Seasonal and annualized nutrient requirements. While daily nutrient requirement functions are both valuable and customary, commercial cow--calf producers do not balance rations on a daily or, often, even monthly basis. An animal's daily requirements are, therefore, not nearly so important in the commercial cow-calf production system as the requirements over somewhat longer periods of time
Time-dependent bioeconomicmodel of beef breeds
339
corresponding to the physiological stages of animal production, the seasons of forage growth, and/or the annual production cycle. Furthermore, because each daily nutrient requirement is itself a function solely of time and certain production parameters, periodic nutrient requirements may be computed by integrating the daily requirement equation over the term (days) of the time period in question. While the time interval may be varied at will, for analyses of the commercial herd breed choice decision an annual interval, reflecting the nutrient requirements of a cow of y years of age as of calving, will suffice. (Annual integrals of the daily nutrient requirement functions are available from the authors on request.)
Herd distribution and aggregation Assuming that animals which die do not wean a calf and, following commercial practices, that any animal which fails to wean a calf is culled, regardless of age, the number of cows of each age (N(i, y)) in a herd of the ith breed group, from replacement (y -- 0) through an economically optimal culling age (y = st) N(i, O) y =0 N(i, y) = [1 - DL(i, y)]N(i, y - 1) y = 1 (18) WR(i, y)N(i, y - 1) 2 < y < si and the total herd size (N(i, .)) is the sum of animals in all age groups. Hence, the age distribution of a herd is fully defined by the age specific productivity values associated with culling (WR(i, y) and DL(i, y)) and the optimal culling age (si) by breed group. For notational convenience this age distribution is represented by the probability distribution function si
D,(y) = N(i, y ) / ~ N(i, y) = I~(DL(i, 1), WR(i, y), si) y=b
as shown in Fig. 1 for average weaning and death loss rates across breeds through s; = 15 years of cow age. The optimal culling age (st) required to define llt(y) for each breed group is dependent upon economic conditions reflecting resource availability and prices. However, once determined, this function may be used to compute the average daily nutrient requirements of the total herd as the sum of the nutrient requirements for each age of cow weighted according to the distribution function. Economic considerations in commercial production Economic models of commercial cow evaluation, selection, and replacement "decisions have traditionally assumed an objective of maximizing the
340
B. E. Melton, IV. A. Colette, K. J. Smith, R. L. Willham 14~ 0
129
©
i0~
.o
8%
0
i
6%:
0
4%! 2% 0% 0
Fig. 1.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Y e a r s of Age Average herd distribution, by age, through 15 years.
net present value (NPV) of the animal or herd of animals over an infinite (or at least multi-generational) life (Melton, 1980; Melton & Colette, 1993). This objective may be expressed as maximizing the NPV of an infinite series of replacements of the ith breed (C(i, b, si, oo)), when each is acquired at weaning age and optimally culled at st based upon the NPV of the first generation in the series (C(i, b, st, 1)), i.e. max C(i, b, st, "~) = si
C(i, b, s t, l) 1 - (1 + p) (s~b)(1 + y) (srb)
(19)
si NRi(y) M,(si) C(i, b, st, 1) = • + ~:b÷~ (1 + p)(~) (1 + p)(~)
Mr(b)
The derivation and implications of this objective, as well as solution criteria with respect to an economically optimal culling age under fixed prices, are presented by Perrin (1972) and Melton (1980). Unlike prior applications, however, in the case at hand one must not only determine an optimal culling age for an average cow, but the breed group that maximizes the NPV of a herd of N(i, .) such animals distributed with respect to age through a breed-specific culling age of s t, recognizing the limited availability of forage nutrients. In other words, the decision variables are (1) the optimal culling age, by breed, and (2) the breed which, when optimally culled, maximizes the NPV of the fixed grazing resource. Under fixed relative prices, a single culling age will be optimal for each breed group. Therefore, a single optimal herd distribution exists for each breed group under each relative price ratio. Furthermore, the amount of grazing nutrients available are proportional to the grazing land area for
Time-dependent bioeconomic model of beef breeds
341
a given forage system. Ignoring random variations in forage production due to climatic variation, the problem may thus be expressed as one of maximizing the annual returns per acre due to grazing by the constant sized cow herd, when that herd is optimally culled and stocked relative to the forage nutrients available, or max {max (¢ri[f~(si) + llT~(si) - ~.(b)])} i
(20)
si
where
[~.(Si) = ~, ai(y)NRi(Y) Y s~l
~-(s;) = 2 [1 - WR(i, y) - DL(i, y)]l)~(y)M~(y) y=2
+ [1 - DL(i, si)]~-~i(si)Mi(si) = Y
ME* Y~ lIi(y)[ME(i, y) + WR(i, y)me(i, y)] o'~ = min
Y
or DP* ~, f~i(y)[DP(i, y) = WR(i, y)dp(i, Y)I Y
is the expected annual stocking rate (head per acre) such that the number of average animals in a fixed herd size is just adequate to fully utilize the most limiting nutrient produced per unit land area (ME* or DP*) and all replacements are reared (b = 0) and culled only because of death loss prior to initial calving age (y = 2). The breed group with the greatest value of eqn (20) will be that group which, when optimally culled (the inside maximization) under fixed price ratios and nutrient yields, maximizes the returns to the fixed land area and grazing resources over time (the outside maximization). Assuming that choice of breed group does not differentially affect land market values, this result also maximizes the NPV of the beef production enterprise on the fixed land area; a criterion more nearly representative of prevailing commercial production conditions and practices. From the necessary (first-order) condition for maximization of the inside maximand of eqn (20), the following criteria can be derived with respect to the optimal culling and replacement age: zx[,g(s,) +
(s3 -
,Xo-
+
(si) -
(21)
342
B. E. Melton, W. A. Colette, K. J. Smith, R. L. Willham
Animals should therefore be culled, on the basis of age, when the negative percentage change in stocking rate due to culling them (the right-hand-side of eqn (21)) is equal to the percentage change in average (herd) returns per unit land area (the left-hand-side of eqn (21)), including consideration of the net replacement costs. Because animal weights, and thus nutrients required, increase with age, the percentage change in stocking rate is expected to be negative with respect to increased culling age until maturity when declining productivity (in terms of fewer calves per cow) may cause stocking rates to increase slightly at later culling ages. Conversely, beef cows do not typically reach their maximum productivity until about 7-8 years of age. Hence, the percentage change in returns per acre is expected to increase to the age of maturity, then decline slightly. One could proceed to solve for the optimal culling age according to the criteria specified by eqn (21) with respect to each breed group, then select the breed group (from amongst the finite number available) with the greatest value of eqn (20). However, with discrete culling and breed choice decisions, no computational advantage would be gained by this procedure. Specifically, the denominators of eqn (21) represent the same computations as eqn (20). Thus, as a practical matter, one can simply solve eqn (20) directly to determine the culling age, by breed, which maximizes net returns per acre; then select the breed with the greatest returns from amongst those available.
MODEL APPLICATION ILLUSTRATED: WEST TEXAS COMMERCIAL COW-CALF PRODUCTION To illustrate the application of this model, a 5000 acre (2023.5 ha) ranch, typical of the Panhandle range areas of Eastern New Mexico and West Texas, was defined. Based upon SCS (USDA/SCS, 1970) values, soil types, forages, and annual forage dry matter, yields were defined for this ranch and converted to annual average yields of M E and D P using seasonally (stage of maturity) adjusted NRC (1984) nutrient concentrations for the representative forages. The representative annual average M E and D P values per acre used were 375 Mcal and 8.8 kg, respectively. Breed productivity estimates with respect to the parameters of growth (Jenkins et al., 1991), lactation (Green et al., 1991) and weaning rate (Cundiff et al., 1986) were derived from the GPE study for 16 breed groups representing the cross of a sire breed with average Hereford and Angus cows according to the procedures described by Melton et al. (1993). These values are summarized in Table 2. Because differential
Time-dependent bioeconomic model o f beef breeds
343
TABLE 2 Production Parameters of Alternative Crossbred Cow Groups a Sire breed
Ai
Bi
IO00(K i)
Ci
Angus Brown Swiss Brahman Charolais Chianina Gelbvieh Hereford Jersey Limousin Maine Anjou Pinzgauer Red Poll Sahiwal Simmental South Devon Tarentaise
512 520 550 554 589 539 500 425 516 583 525 511 486 514 512 519
478 482 513 517 550 501 467 397 482 544 489 476 452 477 477 485
1-745 1-772 1.976 1.615 1.516 1.711 1.944 0-977 1.616 1.581 2.139 1.581 2.009 1.843 1.745 2.043
3.199 1.866 2.614 3.548 3.199 1.866 3.199 1.683 3.548 3.049 1.936 2.602 2.798 1.866 3.002 1.960
lO0(l i) 1.660 1.870 1.480 1.660 1.660 1.870 1-660 1.870 1.660 1.660 1.870 1.660 1.480 1.870 1.660 1.870
t*
0i
WR(i, 7)
285-5 286.0 289.4 287.0 287.3 286.7 285.5 285-0 288.1 286.2 286.5 286.1 290.5 287.2 286.9 287.1
0.012 35 0.012 72 0.012 48 0.012 58 0.012 75 0.012 69 0-012 25 0.011 69 0.012 24 0.012 80 0.012 51 0.01243 0.012 14 0.012 58 0.012 39 0.012 28
0.84 0-85 0.86 0.80 0.86 0.87 0.84 0.84 0.82 0.86 0.85 0.79 0-89 0-83 0.85 0.85
a Values derived from Jenkins et al. (Ai, B i and K3; Green et al. (Ci and O; and Cundiff et al. (t* and WR(i, 7)). 0t solved from parameters as shown in eqn (11).
death loss rates were unavailable, a constant death loss across breed groups at the average specified by eqn (13) was assumed. Annual herd nutrient requirements were computed as previously described for each alternative culling age (2-15 years). These values were then used to determine the maximum stocking rate for each possible culling age, as shown in Fig. 2 for selected breed groups. Relative prices per unit weight for weaned steers, heifers, and culled cows were $1.3779, $1.1574, and $.6600 per kg, respectively. Based upon an examination of over 30 years of historical price data applicable in the area, these prices were judged to reflect the relative prices producers selling in the fall, at weaning, might reasonably expect to receive over the long-run. The implications of alternative relative price ratios were then examined by changing cull cow prices by +$. 16 per kg while calf prices are held constant. Optimal stocking rates, culling ages, and returns per acre are summarized in Table 3 for each breed group and alternative price ratio defined in terms of the ratio of cull cow to calf prices. These results demonstrate a marked economic preference for smaller, higher productivity animals for the resource base defined. Sahiwal, which are optimal among the breed groups considered for all price ratios, possess these characteristics (small mature weight, fast growth, high
344
B. E. Melton, IV. A. Colette, K. J. Smith, R. L. Willham
6.0
-~%~.~ y~k [ ,..,~.. ~,-,~.,,vAngus Brahman ~ PTnzgauer ~ Sahiwal~Chianin--a_ JerseyT:]/ 0 0
x
5.5
5.0
cn 4.5 C
~
4.0
.
3.5
5.0
I
I
I
I
I
I
I
i
L
I
2
5
4
5
6
7
8
9
10
11
12
I
I
I
15
14
15
CuLling Age Fig. 2.
Stocking rate (× 100) for selected breed groups by culling age.
TABLE 3 Optimal Culling Age (si) , Stocking Rate (ori x 100), and Returns per Acre (S/A) for Alternative Breed Groups and Price Ratios Breed group
Angus Brown Swiss Brahman Charolais Chianina Gelbvieh Hereford Jersey Limousin Maine Anjou Pinzgauer Red Poll Sahiwal Simmental South Devon Tarentaise
L o w price ratio
Avg. price ratio
Si
O"i ;< 100
$/A
si
10 10 10 a 9 9 11 10 10 9 11 " 10 11 10 11
3.99 3-82 3.40
5.067 5.120 5-167
3.59 3.71 3.91 4-66 4.15 3.55 3.49
5-025 5.208 5.139 5.240 4.907 5.047 5-194
3.87 3-84 3.96 3.61
5-597 4-988 5.146 5.161
8 8 9 8 7 8 9 9 8 7 9 8 9 9 8 9
~ X 100 S / A
4.04 3-86 3.42 3.88 3-66 3.74 3.93 4.68 4.19 3-63 3-51 4.30 3.89 3.86 4.01 3.63
5.533 5.558 5-564 5-243 5.488 5-628 5-582 5.690 5.422 5.497 5.594 5-251 5-961 5.447 5.597 5.572
High price ratio si
6 6 7 6 5 6 7 7 6 5 7 6 7 6 6 7
o"i X 100 S/A
4.14 3.97 3-48 3-97 3.81 3.85 4.00 4.76 4.30 3.77 3.57 4.40 3-97 3.98 4.12 3.70
6-058 6-052 6.003 5.830 6.038 6.113 6.068 6.180 5.999 6-028 6.024 5.860 6.377 5.952 6.108 6.021
"Not optimal at any alternative culling age considered by second-order conditions for a maximum. One would, therefore, retain in the herd for the longest possible time (15 years in this example) before culling.
Time-dependent bioeconomic model of beef breeds
345
milking ability, and high weaning rate). Conversely, large, slow maturing breeds with marginal reproductive capacity, such as Charolais, are least profitable for the fixed resource area defined. These results would seem to contradict many recent trends which appear to have attached a premium to larger size. One can only speculate that the observed size premium has been an over-reaction on the part of many producers to the dwarfism introduced into commercial US beef herds through show-rings in the 1950s. An increase in cull cow prices relative to calf prices results in cows being culled at an earlier age. However, these results also indicate that an increase in relative cull cow prices causes an increase in stocking rate to fully utilize the available forage. Thus, for a given forage availability the total cow herd may actually be expected to increase in response to an increase in cull cow prices or, conversely, to increase as calf prices decline relative to cull cow prices. This peculiar economic result may be attributed to the fact that as culling age is reduced in response to higher cull cow prices, the herd utilizing the fixed forage resource is increasingly young, with lower per head nutrient requirements, allowing more animals to be supported by the same quantity of forage resources.
IMPLICATIONS There are obvious shortcomings in the model presented in this paper. Most notable of these is the fact that all evaluations are conducted at the breed group means. While these means may provide an initial indication of comparative advantage, in practice, choices are also made between individuals within breed groups and among production practices which may cause animals to deviate from their genetic potential. For example, animals may not sustain the same relative condition as they grow; losing weight at some times and growing more rapidly at others. However, the modifications needed to address these issues can be accommodated easily within the framework of the model by (1) altering parameters to reflect individual performance, (2) introducing stochastic elements to capture random performance variance by individuals, and/or (3) imposing management restrictions which preclude animals achieving their genetically defined performance. In each case animal performance can still be reduced to a relatively simple time-dependent bioeconomic model of alternative breeds, individuals within breed, and/or production parameters. Despite its limitations, the model developed has substantial advantages beyond its above noted flexibility. These include its explicit recognition
346
B. E. Melton, W. A. Colette, K. J. Smith, R. L. Willham
and incorporation o f the time-dependent nature of cow-calf production, including relationships between alternative parameters defining breed differences. Equally important is the fact that the solution to the model is relatively simple and straight forward. As a result, the model can be readily solved by those with limited modelling background or training in optimization using desktop personal computers. It is, therefore, amenable for use by more sophisticated beef herd managers as well as extension personnel and others providing management input to farm and ranch decision-makers.
ACKNOWLEDGMENTS The Midwest Agribusiness Trade and Research Information Centre provided a portion of the support for this project.
REFERENCES Bentley, E., Waters, J. R. & Shumway, C. R. (1976). Determining optimal replacement age of beef cows in the presence of stochastic elements. South. J. Agri. Econ., 8, 13. Brody, S. (1945). Bioenergetics and Growth. Reinhold Publishing Company, New York. Cundiff, L. V., Gregory, K. E., Koch, R. M. & Dickerson, G. E. (1986). Genetic diversity among cattle breeds and its use to increase beef production efficiency in a temperate environment. Proceedings of the Third Worm Congress on Genetics Applied to Livestock Production, Vol. IX :271, Lincoln, NE USA. Green, R. D., Cundiff, L. V., Dickerson, G. E. & Jenkins, T. G. (1991). Output/input differences among nonpregnant lactating bos indicus-bos taurus and bos taurus-bos taurus cross cows. J. Ani. Sci., 69, 3156. Jenkins, T. G., Kaps, M., Cundiff, L. V. & Ferrell, C. L. (1991). Evaluation of between- and within-breed variation in measures of weight-age relationships. J. Ani. Sei., 69, 3118. Melton, B. E. (1980). Economics of beef cow culling and replacement decisions under genetic progress. West. J. Agri. Econ., 5, 137. Melton, B. E. & Arden Colette, W. (1993). Potential shortcomings of output : input ratios as indicators of economic efficiency in commercial beef breed evaluations. J. Ani. Sci., 71, 579. Melton, B. E., Arden Colette, W. & Smith, K. (1993). Optimal Breed Choices for Commercial Cow-Calf Production in the Texas Panhandle." An Application of Bio-Economic Modelling to Breed Evaluations. Texas Agr. Exp. Sta. Bull. B-1716, College Station, TX, USA. Moe, P. W. & Tyrrell, H. F. (1971). Metabolizable energy requirements of pregnant dairy cows. J. Dairy Sci., 55, 480.
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National Research Council (NRC) (1984). Nutrient Requirements of Beef Cattle, 6th Edn. National Academy Press, Washington, D.C. Neville, W. E., Jr. & McCullough, M. E. (1968). Calculated energy requirements of lactating and non-lactating Hereford cows. J. Ani. Sci., 29, 823. Perrin, R. K. (1972). Asset replacement principles. Am. J. Agri. Econ., 54, 60. Rogers, L. F. (1972). Economics of replacement rates in commercial beef herds. J. Ani. Sci., 34, 921. United States Department of Agriculture-Soil Conservation Service (US-DA/SCS) (1970). Soil Survey of Randall County, Texas. USDA Soil Conservation Service. US Government Printing Office, Washington, DC. Wilcox, C. J., Gaunt, S. N. & Farthing, B. R. (1971). Genetic interrelationships of milk composition and yield. Southern Cooperating Series Bulletin No. 155. Interregional Publication of the Northeast and Southeast State Agricultural Experiment Stations.
i!i,I ELSEVIER
A Time-Dependent Bioeconomic Model of Commercial Beef Breed Choices* Bryan E. Melton Department of Economics and Animal Science, 467 Heady Hall, Iowa State University, Ames, Iowa 50011, USA
W. Arden Colette Department of Agricultural Business and Economics, West Texas State University, Canyon, Texas 79016, USA
Kenneth J. Smith Department of Animal Science, Texas Tech University, Lubbock, Texas 00000, USA
&
Richard L. Willham Department of Animal Science, Iowa State University, Ames, Iowa 50011, USA
(Received 10 May 1993; revised version received 19 July 1993; accepted 20 July 1993)
A BSTRA CT The breed choice decision requires simultaneous consideration of both economic and physical relationships in light of the multi-year, multiproduct nature of commercial cow-calf production. To accomplish this a time-dependent bioeconomie model of cow-calf production was developed in which both physical output-input relationships and production decisions are expressed in terms of cow age and stage of production. The breed choice and optimal herd distribution (culling age and proportion by * Journal series paper #J-15452 of the Iowa Agricultural and Home Economics Experiment Station, Project #3175. 331
332
B. E. Melton, W. A. Colette, K. J. Smith, R. L. Willham age) decisions are then cast within the economic framework o f maximizing net returns per unit of nutrient resource derived from a fixed land area. The resulting bioeconomic model is easily solved, as illustrated by an empirical example evaluating 16 alternative breed choices for a representative West Texas ranch. Results indicate a marked economic preference for smaller, higher productivity animals, such as Sahiwal. Conversely, large, slow maturing breeds with marginal reproductive capacity, such as Charolais, are least profitable for the fixed resource area defined.
INTRODUCTION In the last 15 years the number of beef breeds available to commercial producers in the US has increased many fold. Producers are, therefore no longer restricted to Hereford, Angus, or animals of similar genetic composition in their breed choice decisions. They may now select from an array of breeds representing an unprecedented range of biological types and genetic abilities. While such diversity enhances the flexibility of commercial producers to adapt to changing market and economic conditions, it also complicates the breed choice decision. Commercial producers, marketing primarily weaned feeder calves, must now make breed choice decisions that balance the current and long-term productivity of a potentially diverse cow herd against limited production resources, such as land area, in light of prevailing and expected economic and market conditions. Such decisions require simultaneous consideration of multiple facets of the cow-calf production process. The most notable of these, in addition to resources available (especially with respect to forage nutrient production), market prices, and prevailing economic conditions, are the following: (1) the multi-product, multi-year nature of beef cows in which each cow in the herd is expected to produce a weaned calf in each year of her economic life as well as becoming an additional marketable product upon culling; (2) differences between breeds in annual reproductive (calving rate and weight) and productive (weaning rate and weight) performance; and (3) differences between breeds in nutrient requirements for production and reproduction including the cow's own weight maintenance and growth requirements. In this paper a time-dependent breed evaluation model is developed relecting both physical and economic considerations in commercial breed choice. Applications of this model to the breed choice decision are then illustrated for conditions representative of commercial cow-calf production in the range areas of the West Texas Panhandle.
Time-dependent bioeconomic model of beef breeds
333
M O D E L SPECIFICATION Cow herd nutrient requirements, by stage of production, are central to breed choice decisions. Larger animals tend to produce larger progeny, but have simultaneously greater nutrient requirements for maintenance, growth, and production. Hence, for a fixed land area, implying fixed levels of forage nutrients available, producers must balance animal size and per head nutrient requirements against animal numbers and herd size. The variables required to reflect these factors in the model include various measures of animal performance, weight, and nutrient requirements, as well as variables reflecting the prevailing economic and resource conditions. In defining the variables for weight, weight gain, and nutrient requirements, the convention of using upper case letters to refer to the parent (cow) or herd and lower case to refer to the progeny (calf) is adopted. Furthermore, an additional variable, y, is added to calf variable names whenever necessary to reflect the age, in years, of the calf's dam. Additional variable naming conventions include the use of an over-bar (-) to represent aggregate herd averages of the variable across animal ages (in years), a hat (^) to represent adjusted values of the variable in question, dot ( ' ) notation replacing the breed component of the variable name to denote averages across breeds, and substitution of years (y) for days (t) in the variable name to indicate the summation (integration) of daily values to an annual basis. In light of the large number of variables and variations involved in the development of the model, only root variable definitions for the individual cow are summarized in Table 1.
Animal nutrient requirements Energy and protein are the primary nutrients of concern under range grazing conditions. A cow's daily nutrient requirement for M E may be written as the sum of her requirements for each biological function performed: maintenance (m), growth (g), lactation (/), and reproduction (p). Thus ME(i, t) = MEre(i, t) + MEg(i, t) + MEl(i, t) + MEp(i, t) (1)
Cow daily nutrient requirements Various estimates of these daily nutrient requirements, by biological function, exist in the literature of animal science.. Among these are Neville & McCullough (1968), who have estimated the daily M E requirement for maintenance and growth (weight gain) of post-pubescent cows in temperature neutral environments to be MEre(i, t) + MEg(i, t) = 0"1374W(i, 00.75 + 6"3G(i, t)
(2)
334
B. E. Melton, W. A. Colette, K. J. Smith, R. L. Willham
TABLE 1 Variable definition
Variable
Units
Definition
ME(i, t)
mcal/day
ME,..g,t.p(i, t)
mcal/day
DP(i, t)
kg/day
DPm.g,l,p(i, t)
kg/day
W(i, t) G(i, t) Ai Bi
kg kg/day kg kg
Total daily metabolizable energy (ME) required by an animal of the ith breed on day of age, t. Daily M E required for maintenance (m), growth and weight gain (g), lactation (l), or reproduction and pregnancy (p), respectively, by an animal of the ith breed on day of age, t. Total daily digestible protein (DP) required by an animal of the ith breed on day of age, t. Daily DP required for maintenance (m), growth and weight gain (g), lactation (/), or reproduction and pregnancy (p), respectively, by an animal of the ith breed on day of age, t. Weight of an animal of the ith breed on the tth day of age. Weight gain of an animal of the ith breed on the tth day of age. Asymptotic mature weight of an animal of the ith breed. Asymptotic total weight gain of an animal of the ith breed. Growth curve rate of growth parameter of an animal of the ith breed. Weight of milk produced by a cow of ith breed on the tth day of lactation. Parameter of daily lactation of an animal of the ith breed. Parameter of the rate of change in daily lactation of an animal of the ith bread. Prenatal weight of an embryo of the ith breed on the tth day of gestation (p). The prenatal rate of growth for an animal of the ith breed. Percentage death loss of an animal of the ith breed that is y years of age as of calving. Percentage weaning rate of an animal of the ith breed that is y years of age as of calving. Age adjustment factor (proportion of maturity) for an animal of the ith breed that is y years of age as of calving. Number of cows of the ith breed and yth age in years, as of calving, in a herd. The probability distribution function of animal age (proportion of animals in herd by years of age) for the ith breed. Optimal culling age, in years, for the ith breed. The net present value of the nth generation of an animal of the ith breed when each successive generation is acquired at b years of age and culled at s i years. The net annual returns of an animal of the ith breed at y years of age, as of calving. The market value of an animal of the ith breed at y years of age, as of calving. The applicable discount rate. The applicable rate of (exogenous or endogenous) genetic progress due to selection. Average annual returns per unit land area for the ith breed when optimally culled and stocked. Total ME provided per unit land area. Total DP provided per unit land area. The stocking rate of the ith breed when optimally culled and land resources are fully utilized.
Ki
M(i ,tt)
kg/day
Ci
kg
li w(i, tp)
kg
0i DL(i, y)
%
WR(i, y)
%
AA(i, y) N(i, y)
~i(Y) si C(i, b, si, n)
Y $
NRi(Y)
$
M(y)
$
P Y
R(s3
$
ME*
mcal/ha kg/ha hd/ha
DP* ~ri
Time-dependent bioeconomic model o f beef breeds
335
The additional daily M E required for lactation has also been estimated (Neville & McCullough, 1968) as
MEl(i, t)
~0.41W(i, t)°75+ 1.9G(i, t)+O.7433M(i, tt) 0
/
(3)
0 tt > tt* where the total lactation cycle is tt* days in length. ,Reproductive M E requirements for prenatal growth are similarly estimated (Moe & Tyrrell, 1971) as
O
j0.000 567 exp(O.O174tp)W(i, t I tp=0) °75 / 0
W(i, t) + A i - B i exp ( - - K i t )
(5)
where
d W(i, t) G(i, t) - - dt
-
KiB i
exp ( - Ki t)
(6)
Furthermore, M(L tt) may also be expressed in terms of a lactation cycle function (Jenkins et al., 1991)
M(i, tl) -
Ii
6",.exp (lttl)
(7)
Substituting eqns (5) to (7) into eqn (1) allows the total daily M E function for post-pubescent cows to be reduced to a function of only time: day of age (t), day of lactation (it), and day of gestation (tp). Differences in the estimated daily M E requirements between cows, or breeds of cows, of comparable age and production status are then represented directly by differences in a limited number of breed-dependent parameters (At, Bt, Ki, C t and 1~) associated with estimated growth curves and lactation cycle functions (eqns (5) and (7)). Maintenance protein losses are dominated by fecal loss (approximately 80%) according to N R C (1984) recommendations. Given a fecal protein loss of 3.34% of dry matter intake and assuming a maximum average
336
B. E. Melton, W. A. Colette, K. J. Smith, R. L. Willham
forage dry matter intake of 2% of body weight, the DP requirement for maintenance is approximately 0.000 84 times body weight. DP requirements for weight gain and milk production are taken directly from NRC (1984, p. 9) where the weight gain requirement function reported by NRC is one adapted from Fox. However, NRC (1984) recommendations for 55 g of DP per day in the last trimester of pregnancy are re-specified to a continuous function of average fetal weight over the last trimester, resulting in an average requirement of 0-0023 kg per kg of fetal weight and an average third trimester fetal weight of approximately 24 kg. Estimates of total daily protein requirements can, therefore, be derived in terms similar to those used in the ME functions as follows: DPm(i, t) + DPg(i, t) = 0"000 84 W(i, t) + G(i, t) (8) [0"235 - 0"000 26W(i, t)]
DPt(i, t) = { 0.0335M(i,0 tl)
0 < t/ - tt*
(9)
t l < tff
DPp(i, t) -- {0-0023w(i,0 tp)
O
(10)
Assuming that progeny weight on any day of gestation can be expressed as an exponential function as weight increases from zero at conception to birth weight (A i - B), w(i, tp) = exp (O~tp) (11) where 0i = In (At - B~)/t* also allows the DP(i, t) functions to be expressed in terms of time and the parameters of growth and lactation.
Calf daily nutrient requirements While progeny may receive the majority of their nutrient requirements from the cow while nursing, over the course of a typical lactation cycle the calf will frequently acquire additional nutrients. Similarly, calves to be retained for replacement in the breeding herd have nutrient requirements from weaning to puberty which must also be fulfilled. NRC (1984) recommendations suggest that eqn (8) is adequate for calf protein requirements, but that a higher energy level is required than would be provided by eqn (2). Accordingly, calf daily ME requirements are estimated on a female basis from NRC (1984) recommendations as
me(i, t) = -29.3039 + 0.3888w(i, t) °7s + 41-7419g(i, t) (0.004 34) (0.002 21)
-10-8340g(i, 0-094 84
t) 2
(0.510 30) r 2 = 0.9999
(12)
Time-dependent bioeeonomicmodel of beef breeds
337
for t < 600, an arbitrarily selected age which should exceed the onset of puberty (standard errors are shown in parentheses below the estimated regression coefficients). Because males tend to grow faster and to a larger size than females, additional nutrient adjustments are required for sex of progeny. N R C (1984) values suggest that this adjustment should be plus 12% for preweaning (intact) male calves. (For simplicity, no males are assumed to be retained after weaning as replacement sires are acquired exogenously in most commercial herds.) While nursing, the progeny's requirements for both ME and DP must then be reduced by the nutrient yield of the dam's milk, obtained by multiplying eqn (7) by 0.564 and 0.030, respectively (NRC, 1984); subject to the conditions that the result is non-negative.
Adjustments to cow--calf-daily nutrient requirement functions Additional adjustments to the daily nutrient requirements are needed to reflect that not all cows will wean a calf each year; that some cows will die; and that performance, with respect to both these characteristics as well as production parameters, such as calf weight and milk production, will be influenced by cow age. Annual death losses and weaning rates, in each year of an average cow's life (y), without respect to breed, are estimated from data developed by Rogers (1972) and Bentley et al. (1976), respectively, for cows 2 to 15 years of age as follows: /-2.4078 +0.32269y~ DL(.,y < 10)xl00 -- 2 + exp 1` (0-2155) (0.023 73)]
r2=0.9585
(13a)
[ 6 . 4 0 0 2 - 0.55421y] DL(.,y< 10)xl00 = 7 - exp 1,(0.161 34) (0.02373)]
r2=0.9672
(13b)
WR(.,y< 10)xl00= 74.9647 + 6.6532y - 0.586 01y2 r2=0.9956 (0.74006) (0.31010) (0.02206)
(14)
and
Breed differences for weaning and death loss rates, by age of dam, are generally not reported. Hence, the adjustment required to obtain these breed-specific values over time must be based on point in time estimates relative to the above equations. For example, weaning rates for alternative breeds may be reported on an age constant basis (typically seven years of age). Equation (14) may then be used to estimate breed-specific weaning rate differences at any age based upon the ratio of reported point in time (y*) weaning rates for the ith breed (WR(i, y*)) to the predicted weaning rate from eqn (14) at the same age (WR(., y*)), i.e.
wR(i,
WR(i, y) = ~ . ,
'ly*)
~ WR(., y) y'))
338
B. E. Melton, IV. A. Colette, K. J. Smith, R. L. Willham
Multiplicative age of dam adjustments, as a percent of mature production, are estimated using monthly additive adjustments reported by Wilcox et al. (1971)
AA(., y) = 0.4959 + 0.1649y - 0.0164y 2 + 0.00046y 3 r 2 = 0.9909 (0-00796) (0.00834) (0.001 10) (0.00004)
(15)
These relationships are used to adjust both M E and DP requirements. For purposes of illustration, the expected daily M E requirements for a cow of age y, adjusted for age, weaning rate, and death loss assuming a uniform distribution, are shown below:
M^E(i, t) = {1 - [t - 365y] DL(i, Y~)}365 x { MEre(i, t) + MEg(i, i) + WR(i, y) × [MEt(i, t) + MEp(i, t)]}
(16)
where
MEt(i,t)= ~O.O41W(i, t)°75 + l.9G(i, t)+ AA(i,y)O.7443M(i, tl)
O
0
tl > t~
and
MEp(i, t)= ~AA(i, y)0.000567 exp(O.O174tp)W(i, t [ tp = 0)0.75 0 < tp
o
tp>¢
The corresponding net (of milk production) daily pre-weaning M E requirements on the ith day of age for the progeny of a cow y years of age are similarly adjusted to obtain expected daily values as follows:
me(i, t, y) - AA(i, y) 0.569M(i, t) = -29.3039 + 0.3888[AA(i, y)w(i, t)] °75 + 41-7419g(i, t) (0.004 34) (0.002 21) (0.150 30) -10.8340g(i, t) 2 - AA(i, y)O.569M(i, tl) (0.09484)
(17)
which are expressed on a per cow basis when multiplied by WR(L y). Comparable expectations may then be computed for DP.
Seasonal and annualized nutrient requirements. While daily nutrient requirement functions are both valuable and customary, commercial cow--calf producers do not balance rations on a daily or, often, even monthly basis. An animal's daily requirements are, therefore, not nearly so important in the commercial cow-calf production system as the requirements over somewhat longer periods of time
Time-dependent bioeconomicmodel of beef breeds
339
corresponding to the physiological stages of animal production, the seasons of forage growth, and/or the annual production cycle. Furthermore, because each daily nutrient requirement is itself a function solely of time and certain production parameters, periodic nutrient requirements may be computed by integrating the daily requirement equation over the term (days) of the time period in question. While the time interval may be varied at will, for analyses of the commercial herd breed choice decision an annual interval, reflecting the nutrient requirements of a cow of y years of age as of calving, will suffice. (Annual integrals of the daily nutrient requirement functions are available from the authors on request.)
Herd distribution and aggregation Assuming that animals which die do not wean a calf and, following commercial practices, that any animal which fails to wean a calf is culled, regardless of age, the number of cows of each age (N(i, y)) in a herd of the ith breed group, from replacement (y -- 0) through an economically optimal culling age (y = st) N(i, O) y =0 N(i, y) = [1 - DL(i, y)]N(i, y - 1) y = 1 (18) WR(i, y)N(i, y - 1) 2 < y < si and the total herd size (N(i, .)) is the sum of animals in all age groups. Hence, the age distribution of a herd is fully defined by the age specific productivity values associated with culling (WR(i, y) and DL(i, y)) and the optimal culling age (si) by breed group. For notational convenience this age distribution is represented by the probability distribution function si
D,(y) = N(i, y ) / ~ N(i, y) = I~(DL(i, 1), WR(i, y), si) y=b
as shown in Fig. 1 for average weaning and death loss rates across breeds through s; = 15 years of cow age. The optimal culling age (st) required to define llt(y) for each breed group is dependent upon economic conditions reflecting resource availability and prices. However, once determined, this function may be used to compute the average daily nutrient requirements of the total herd as the sum of the nutrient requirements for each age of cow weighted according to the distribution function. Economic considerations in commercial production Economic models of commercial cow evaluation, selection, and replacement "decisions have traditionally assumed an objective of maximizing the
340
B. E. Melton, IV. A. Colette, K. J. Smith, R. L. Willham 14~ 0
129
©
i0~
.o
8%
0
i
6%:
0
4%! 2% 0% 0
Fig. 1.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Y e a r s of Age Average herd distribution, by age, through 15 years.
net present value (NPV) of the animal or herd of animals over an infinite (or at least multi-generational) life (Melton, 1980; Melton & Colette, 1993). This objective may be expressed as maximizing the NPV of an infinite series of replacements of the ith breed (C(i, b, si, oo)), when each is acquired at weaning age and optimally culled at st based upon the NPV of the first generation in the series (C(i, b, st, 1)), i.e. max C(i, b, st, "~) = si
C(i, b, s t, l) 1 - (1 + p) (s~b)(1 + y) (srb)
(19)
si NRi(y) M,(si) C(i, b, st, 1) = • + ~:b÷~ (1 + p)(~) (1 + p)(~)
Mr(b)
The derivation and implications of this objective, as well as solution criteria with respect to an economically optimal culling age under fixed prices, are presented by Perrin (1972) and Melton (1980). Unlike prior applications, however, in the case at hand one must not only determine an optimal culling age for an average cow, but the breed group that maximizes the NPV of a herd of N(i, .) such animals distributed with respect to age through a breed-specific culling age of s t, recognizing the limited availability of forage nutrients. In other words, the decision variables are (1) the optimal culling age, by breed, and (2) the breed which, when optimally culled, maximizes the NPV of the fixed grazing resource. Under fixed relative prices, a single culling age will be optimal for each breed group. Therefore, a single optimal herd distribution exists for each breed group under each relative price ratio. Furthermore, the amount of grazing nutrients available are proportional to the grazing land area for
Time-dependent bioeconomic model of beef breeds
341
a given forage system. Ignoring random variations in forage production due to climatic variation, the problem may thus be expressed as one of maximizing the annual returns per acre due to grazing by the constant sized cow herd, when that herd is optimally culled and stocked relative to the forage nutrients available, or max {max (¢ri[f~(si) + llT~(si) - ~.(b)])} i
(20)
si
where
[~.(Si) = ~, ai(y)NRi(Y) Y s~l
~-(s;) = 2 [1 - WR(i, y) - DL(i, y)]l)~(y)M~(y) y=2
+ [1 - DL(i, si)]~-~i(si)Mi(si) = Y
ME* Y~ lIi(y)[ME(i, y) + WR(i, y)me(i, y)] o'~ = min
Y
or DP* ~, f~i(y)[DP(i, y) = WR(i, y)dp(i, Y)I Y
is the expected annual stocking rate (head per acre) such that the number of average animals in a fixed herd size is just adequate to fully utilize the most limiting nutrient produced per unit land area (ME* or DP*) and all replacements are reared (b = 0) and culled only because of death loss prior to initial calving age (y = 2). The breed group with the greatest value of eqn (20) will be that group which, when optimally culled (the inside maximization) under fixed price ratios and nutrient yields, maximizes the returns to the fixed land area and grazing resources over time (the outside maximization). Assuming that choice of breed group does not differentially affect land market values, this result also maximizes the NPV of the beef production enterprise on the fixed land area; a criterion more nearly representative of prevailing commercial production conditions and practices. From the necessary (first-order) condition for maximization of the inside maximand of eqn (20), the following criteria can be derived with respect to the optimal culling and replacement age: zx[,g(s,) +
(s3 -
,Xo-
+
(si) -
(21)
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B. E. Melton, W. A. Colette, K. J. Smith, R. L. Willham
Animals should therefore be culled, on the basis of age, when the negative percentage change in stocking rate due to culling them (the right-hand-side of eqn (21)) is equal to the percentage change in average (herd) returns per unit land area (the left-hand-side of eqn (21)), including consideration of the net replacement costs. Because animal weights, and thus nutrients required, increase with age, the percentage change in stocking rate is expected to be negative with respect to increased culling age until maturity when declining productivity (in terms of fewer calves per cow) may cause stocking rates to increase slightly at later culling ages. Conversely, beef cows do not typically reach their maximum productivity until about 7-8 years of age. Hence, the percentage change in returns per acre is expected to increase to the age of maturity, then decline slightly. One could proceed to solve for the optimal culling age according to the criteria specified by eqn (21) with respect to each breed group, then select the breed group (from amongst the finite number available) with the greatest value of eqn (20). However, with discrete culling and breed choice decisions, no computational advantage would be gained by this procedure. Specifically, the denominators of eqn (21) represent the same computations as eqn (20). Thus, as a practical matter, one can simply solve eqn (20) directly to determine the culling age, by breed, which maximizes net returns per acre; then select the breed with the greatest returns from amongst those available.
MODEL APPLICATION ILLUSTRATED: WEST TEXAS COMMERCIAL COW-CALF PRODUCTION To illustrate the application of this model, a 5000 acre (2023.5 ha) ranch, typical of the Panhandle range areas of Eastern New Mexico and West Texas, was defined. Based upon SCS (USDA/SCS, 1970) values, soil types, forages, and annual forage dry matter, yields were defined for this ranch and converted to annual average yields of M E and D P using seasonally (stage of maturity) adjusted NRC (1984) nutrient concentrations for the representative forages. The representative annual average M E and D P values per acre used were 375 Mcal and 8.8 kg, respectively. Breed productivity estimates with respect to the parameters of growth (Jenkins et al., 1991), lactation (Green et al., 1991) and weaning rate (Cundiff et al., 1986) were derived from the GPE study for 16 breed groups representing the cross of a sire breed with average Hereford and Angus cows according to the procedures described by Melton et al. (1993). These values are summarized in Table 2. Because differential
Time-dependent bioeconomic model o f beef breeds
343
TABLE 2 Production Parameters of Alternative Crossbred Cow Groups a Sire breed
Ai
Bi
IO00(K i)
Ci
Angus Brown Swiss Brahman Charolais Chianina Gelbvieh Hereford Jersey Limousin Maine Anjou Pinzgauer Red Poll Sahiwal Simmental South Devon Tarentaise
512 520 550 554 589 539 500 425 516 583 525 511 486 514 512 519
478 482 513 517 550 501 467 397 482 544 489 476 452 477 477 485
1-745 1-772 1.976 1.615 1.516 1.711 1.944 0-977 1.616 1.581 2.139 1.581 2.009 1.843 1.745 2.043
3.199 1.866 2.614 3.548 3.199 1.866 3.199 1.683 3.548 3.049 1.936 2.602 2.798 1.866 3.002 1.960
lO0(l i) 1.660 1.870 1.480 1.660 1.660 1.870 1-660 1.870 1.660 1.660 1.870 1.660 1.480 1.870 1.660 1.870
t*
0i
WR(i, 7)
285-5 286.0 289.4 287.0 287.3 286.7 285.5 285-0 288.1 286.2 286.5 286.1 290.5 287.2 286.9 287.1
0.012 35 0.012 72 0.012 48 0.012 58 0.012 75 0.012 69 0-012 25 0.011 69 0.012 24 0.012 80 0.012 51 0.01243 0.012 14 0.012 58 0.012 39 0.012 28
0.84 0-85 0.86 0.80 0.86 0.87 0.84 0.84 0.82 0.86 0.85 0.79 0-89 0-83 0.85 0.85
a Values derived from Jenkins et al. (Ai, B i and K3; Green et al. (Ci and O; and Cundiff et al. (t* and WR(i, 7)). 0t solved from parameters as shown in eqn (11).
death loss rates were unavailable, a constant death loss across breed groups at the average specified by eqn (13) was assumed. Annual herd nutrient requirements were computed as previously described for each alternative culling age (2-15 years). These values were then used to determine the maximum stocking rate for each possible culling age, as shown in Fig. 2 for selected breed groups. Relative prices per unit weight for weaned steers, heifers, and culled cows were $1.3779, $1.1574, and $.6600 per kg, respectively. Based upon an examination of over 30 years of historical price data applicable in the area, these prices were judged to reflect the relative prices producers selling in the fall, at weaning, might reasonably expect to receive over the long-run. The implications of alternative relative price ratios were then examined by changing cull cow prices by +$. 16 per kg while calf prices are held constant. Optimal stocking rates, culling ages, and returns per acre are summarized in Table 3 for each breed group and alternative price ratio defined in terms of the ratio of cull cow to calf prices. These results demonstrate a marked economic preference for smaller, higher productivity animals for the resource base defined. Sahiwal, which are optimal among the breed groups considered for all price ratios, possess these characteristics (small mature weight, fast growth, high
344
B. E. Melton, IV. A. Colette, K. J. Smith, R. L. Willham
6.0
-~%~.~ y~k [ ,..,~.. ~,-,~.,,vAngus Brahman ~ PTnzgauer ~ Sahiwal~Chianin--a_ JerseyT:]/ 0 0
x
5.5
5.0
cn 4.5 C
~
4.0
.
3.5
5.0
I
I
I
I
I
I
I
i
L
I
2
5
4
5
6
7
8
9
10
11
12
I
I
I
15
14
15
CuLling Age Fig. 2.
Stocking rate (× 100) for selected breed groups by culling age.
TABLE 3 Optimal Culling Age (si) , Stocking Rate (ori x 100), and Returns per Acre (S/A) for Alternative Breed Groups and Price Ratios Breed group
Angus Brown Swiss Brahman Charolais Chianina Gelbvieh Hereford Jersey Limousin Maine Anjou Pinzgauer Red Poll Sahiwal Simmental South Devon Tarentaise
L o w price ratio
Avg. price ratio
Si
O"i ;< 100
$/A
si
10 10 10 a 9 9 11 10 10 9 11 " 10 11 10 11
3.99 3-82 3.40
5.067 5.120 5-167
3.59 3.71 3.91 4-66 4.15 3.55 3.49
5-025 5.208 5.139 5.240 4.907 5.047 5-194
3.87 3-84 3.96 3.61
5-597 4-988 5.146 5.161
8 8 9 8 7 8 9 9 8 7 9 8 9 9 8 9
~ X 100 S / A
4.04 3-86 3.42 3.88 3-66 3.74 3.93 4.68 4.19 3-63 3-51 4.30 3.89 3.86 4.01 3.63
5.533 5.558 5-564 5-243 5.488 5-628 5-582 5.690 5.422 5.497 5.594 5-251 5-961 5.447 5.597 5.572
High price ratio si
6 6 7 6 5 6 7 7 6 5 7 6 7 6 6 7
o"i X 100 S/A
4.14 3.97 3-48 3-97 3.81 3.85 4.00 4.76 4.30 3.77 3.57 4.40 3-97 3.98 4.12 3.70
6-058 6-052 6.003 5.830 6.038 6.113 6.068 6.180 5.999 6-028 6.024 5.860 6.377 5.952 6.108 6.021
"Not optimal at any alternative culling age considered by second-order conditions for a maximum. One would, therefore, retain in the herd for the longest possible time (15 years in this example) before culling.
Time-dependent bioeconomic model of beef breeds
345
milking ability, and high weaning rate). Conversely, large, slow maturing breeds with marginal reproductive capacity, such as Charolais, are least profitable for the fixed resource area defined. These results would seem to contradict many recent trends which appear to have attached a premium to larger size. One can only speculate that the observed size premium has been an over-reaction on the part of many producers to the dwarfism introduced into commercial US beef herds through show-rings in the 1950s. An increase in cull cow prices relative to calf prices results in cows being culled at an earlier age. However, these results also indicate that an increase in relative cull cow prices causes an increase in stocking rate to fully utilize the available forage. Thus, for a given forage availability the total cow herd may actually be expected to increase in response to an increase in cull cow prices or, conversely, to increase as calf prices decline relative to cull cow prices. This peculiar economic result may be attributed to the fact that as culling age is reduced in response to higher cull cow prices, the herd utilizing the fixed forage resource is increasingly young, with lower per head nutrient requirements, allowing more animals to be supported by the same quantity of forage resources.
IMPLICATIONS There are obvious shortcomings in the model presented in this paper. Most notable of these is the fact that all evaluations are conducted at the breed group means. While these means may provide an initial indication of comparative advantage, in practice, choices are also made between individuals within breed groups and among production practices which may cause animals to deviate from their genetic potential. For example, animals may not sustain the same relative condition as they grow; losing weight at some times and growing more rapidly at others. However, the modifications needed to address these issues can be accommodated easily within the framework of the model by (1) altering parameters to reflect individual performance, (2) introducing stochastic elements to capture random performance variance by individuals, and/or (3) imposing management restrictions which preclude animals achieving their genetically defined performance. In each case animal performance can still be reduced to a relatively simple time-dependent bioeconomic model of alternative breeds, individuals within breed, and/or production parameters. Despite its limitations, the model developed has substantial advantages beyond its above noted flexibility. These include its explicit recognition
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B. E. Melton, W. A. Colette, K. J. Smith, R. L. Willham
and incorporation o f the time-dependent nature of cow-calf production, including relationships between alternative parameters defining breed differences. Equally important is the fact that the solution to the model is relatively simple and straight forward. As a result, the model can be readily solved by those with limited modelling background or training in optimization using desktop personal computers. It is, therefore, amenable for use by more sophisticated beef herd managers as well as extension personnel and others providing management input to farm and ranch decision-makers.
ACKNOWLEDGMENTS The Midwest Agribusiness Trade and Research Information Centre provided a portion of the support for this project.
REFERENCES Bentley, E., Waters, J. R. & Shumway, C. R. (1976). Determining optimal replacement age of beef cows in the presence of stochastic elements. South. J. Agri. Econ., 8, 13. Brody, S. (1945). Bioenergetics and Growth. Reinhold Publishing Company, New York. Cundiff, L. V., Gregory, K. E., Koch, R. M. & Dickerson, G. E. (1986). Genetic diversity among cattle breeds and its use to increase beef production efficiency in a temperate environment. Proceedings of the Third Worm Congress on Genetics Applied to Livestock Production, Vol. IX :271, Lincoln, NE USA. Green, R. D., Cundiff, L. V., Dickerson, G. E. & Jenkins, T. G. (1991). Output/input differences among nonpregnant lactating bos indicus-bos taurus and bos taurus-bos taurus cross cows. J. Ani. Sci., 69, 3156. Jenkins, T. G., Kaps, M., Cundiff, L. V. & Ferrell, C. L. (1991). Evaluation of between- and within-breed variation in measures of weight-age relationships. J. Ani. Sei., 69, 3118. Melton, B. E. (1980). Economics of beef cow culling and replacement decisions under genetic progress. West. J. Agri. Econ., 5, 137. Melton, B. E. & Arden Colette, W. (1993). Potential shortcomings of output : input ratios as indicators of economic efficiency in commercial beef breed evaluations. J. Ani. Sci., 71, 579. Melton, B. E., Arden Colette, W. & Smith, K. (1993). Optimal Breed Choices for Commercial Cow-Calf Production in the Texas Panhandle." An Application of Bio-Economic Modelling to Breed Evaluations. Texas Agr. Exp. Sta. Bull. B-1716, College Station, TX, USA. Moe, P. W. & Tyrrell, H. F. (1971). Metabolizable energy requirements of pregnant dairy cows. J. Dairy Sci., 55, 480.
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National Research Council (NRC) (1984). Nutrient Requirements of Beef Cattle, 6th Edn. National Academy Press, Washington, D.C. Neville, W. E., Jr. & McCullough, M. E. (1968). Calculated energy requirements of lactating and non-lactating Hereford cows. J. Ani. Sci., 29, 823. Perrin, R. K. (1972). Asset replacement principles. Am. J. Agri. Econ., 54, 60. Rogers, L. F. (1972). Economics of replacement rates in commercial beef herds. J. Ani. Sci., 34, 921. United States Department of Agriculture-Soil Conservation Service (US-DA/SCS) (1970). Soil Survey of Randall County, Texas. USDA Soil Conservation Service. US Government Printing Office, Washington, DC. Wilcox, C. J., Gaunt, S. N. & Farthing, B. R. (1971). Genetic interrelationships of milk composition and yield. Southern Cooperating Series Bulletin No. 155. Interregional Publication of the Northeast and Southeast State Agricultural Experiment Stations.