Economic replacement of a heterogeneous herd.

Production Characteristics and Assumptions

The following discussion provides details concerning the parameters and assumptions that were used in developing this model.

Swine Herd

This analysis uses a stochastic model of swine herd compositional growth that has been estimated as a mixed effects model based on animal feeding trials (Schinckel et al. 2003a). The portion of the model focused on live weight growth is briefly described below as a function of time:

(4) [BW.sub.it]

= (C + [C.sub.i])(1 - exp(-exp (M + [m.sub.i])[t.sup.A])) + b + [e.sub.it]

where [BW.sub.it] denotes the body weight of ith pig at time t; C, M, and A denote fixed population parameters; [c.sub.i] and [m.sub.i] denote the random effects for the ith pig; t denotes the age of the pig; and b denotes body weight at birth (a constant equal to 1.4 kg).

The portion of the model focused on feed consumption is as follows:

(5) [PA.sub.it] = (E + [e.sub.i])[1 - exp([delta] + [delta] [BW.sub.it]

+ [[delta].sub.2] [BW.sup.2].sub.it)] - (E + [e.sub.i])

x [1 - exp([delta] + [[delta].sub.1] BW i,t-1

+ [delta].sub.2] [BW.sup2].sub. i, t-1)]

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(7) [FI.sub.it] = [[PHI].sub.t][BW.sup.[eta].sub.it] + [[phi].sub.it] + [[PSI].sub.t][FA.sub.it]

where [PA.sub.it] denotes the protein accretion for ith pig at time t; [FA.sub.it] denotes fat accretion for ith pig at time t; [FI.sub.it] denotes feed intake for ith pig at time t; E, [[delta].sub.0], [[delta].sub.1], and [[delta].sub.2] denote fixed population parameters for protein accretion; [e.sub.i] denotes a random effect parameter for ith pig's protein accretion (correlated with [C.sub.i]);[[alpha].sub.1], [[alpha].sub.2], [[gamma].sub.1], and [[gamma].sub.2] are parameters for fat accretion; and [[PHI].sub.t], [[phi].sub.t], and [[PSI].sub.t] are parameters for feed intake (these are indexed by t because these constants change with the feed rations to reflect different energy densities).

Because this model focuses on the growth of individual animals in the herd, and reflects the most important determinants of revenue (body weight and the relative proportions of fat and lean) and costs (feed intake), it is ideal for the evaluation of the impacts of heterogeneity on optimal marketing decisions. Details of the biological growth model may be found in Schinckel et al. (2003, 2003a).

Using this model, 100 herds of 1,000 gilts were generated, and body weight and feed intake were recorded daily from the age of 50 to 200 days; this time span is sufficiently large to cover any reasonable cycle length for the grow-finish stage of production. The alternative marketing models were applied to each of the 100 herds.

Production System

The present analysis assumes the use of an AIAO production system. This widely used system accounts for 80% of pigs currently produced in the United States, and has been the focus of work by several authors including Conner and Lowe (2002). Briefly, under this system, all pigs must be marketed and the barn must be cleaned and disinfected before the barn is refilled with young feeder pigs. This does not imply that all pigs must be marketed on the same day, but rather that the barn must be cleaned and disinfected before it is refilled. It is assumed that this cleaning and herd restocking process takes seven days.

This analysis further assumes that the production facilities are a sunk expense, and have no alternative uses. Labor to operate the production facility is subsumed in the production facility cost and is assumed not to vary with either the herd or herd size. As such, both the facility and associated operator labor are considered fixed expenses; the objective to be optimized is thus equivalent to annual returns to these fixed facilities and operator labor. Given low interest rates and the relatively short period it takes a herd to cycle through a barn (less than four months), the discount rate reflecting the time value of money was set to zero.

Production Inputs

Feeder pigs, feed, transportation costs, and a small number of other inputs are explicitly taken into account in calculating the return to fixed facilities and operator labor. Each of these inputs and their costs are detailed below.

Feeder pigs. It was assumed that pigs were purchased from another (nursery) site at fifty days of age. A ten year average (1991-2000) feeder pig price of $42.00 (Li 2003) was used in this analysis.

Feed. As feed is the largest component of cost in the swine grower-finisher production process. To approximate an economically optimal nutrition program, this analysis assumed that pigs would be fed a total of five diets over their growth and finishing phases. The model simulated that animals were fed diets 1-3 during the growth phase, and diets 4-5 during the finishing phase. Feed composition was based on a standard corn-soybean meal diet that included synthetic lysine, a vitamin-mineral premix. Calculation of feed composition and feed cost followed the cost minimization research work by Hill et al. (1998). Decisions concerning the day on which each diet was started were based on Li (2003), and Schinckel (2005). The content for each of these diets is presented in table 1. In addition to feed ingredients, the cost per kg of each diet includes $12/ton for grinding, mixing, and feed transportation (Richert 2005).

Corn. Industry standard nutrient values were used for the corn used in this analysis (yellow grain, National Research Council 1998). Corn prices were calculated by Li (2003) and were based on a ten-year (1991-2003) price average.

Soybean meal Industry standard nutrient values for soybean meal were used in this analysis (dehulled, 48% crude protein soybean meal, National Research Council 1998). The soybean meal price used in this analysis was based upon the same ten-year price average (Li 2003).

Lysine. As dietary lysine is available to hogs through a number of common feeds, producers may mix feed ingredients to obtain the lowest cost of desired lysine levels. Lysine is the first limiting essential amino acid in corn-soybean meal based swine diets (National Research Council 1998). In this instance, corn and soybean meal provided the major sources of lysine and other essential amino acids. Following the Purdue recommended swine diets (Purdue University 2003), it was assumed that 0.15% synthetic Lysine-HCL containing 78% L-Lysine was added throughout the feeding period.

Other feed ingredients. While the relative soybean meal and corn content of diets was adjusted to meet minimum lysine requirements, and varied with the relative prices of these ingredients, other feed ingredients were assumed to be included in fixed quantities. The quantity and composition of other diet ingredients that help ensure that the swine nutrient requirements are met at each growth/finishing stage are presented in table 1 and were based upon the Purdue University recommended standard swine diets (Purdue University 2003).

Transportation. The cost of feeder pig transportation to the production facility is assumed to be included in the feeder pig purchase price. Transportation costs for marketing are based upon ten-year transportation cost averages (1991-2000), and were determined to be $2/head (Li 2003). It is assumed that transportation costs per head are the same whether the entire barn is marketed at once or truckload batches are marketed over time.

In order to most efficiently market a herd, it is assumed that, where possible, animals are shipped in full-truckload batches. Shipping in this manner requires five full truckloads of 170 head/truck, and one partial truckload of the remaining 150 head. It is further assumed that, in shipping, animals are ordered by weight at the time the load is shipped with the heaviest animals shipped first.

Other inputs. For the purposes of this analysis, a variety of other, relatively small, inputs and expenses are included. These expenses are for veterinary services, medication, death loss, and miscellaneous, and the cost of these items are estimated to be $0.09/day (Li 2003).

Marketing

To encourage the delivery of more homogeneous animals, swine processors discount base prices for animals that are outside a desired range of weight and carcass leanness. Hog base price and discount schedules are based on the individual animal's hot carcass weight (head and skin removed) rather than their live weight. This model assumes that the producer has perfect knowledge of the individual pig's live weight and the corresponding hot carcass weight, and it is assumed that the producer can sort animals without error.

Two types of carcass-based payment schemes are commonly used in industry. The first scheme assigns a price ($/cwt) to the animal equal to a base price adjusted by a discount based on live weight. The discount is typically a step function--that is, it is constant over several live weight ranges. The second scheme is more commonly used. Under the second scheme, the price ($/cwt) is again equal to a base price adjusted by a discount (or premium) that is based not only on live weight range, but also on carcass leanness. The present study considers only the second type of payment scheme. A ten-year average live weight price (1991-2000) of $43.00/cwt was used as the base price (Li 2003). It is assumed that the market for live hogs is competitive, that the producer cannot influence the price or discount schedule for either finished hogs or feeder pigs, and that producers face no price risk.

This study makes use of two payment schedules which were established and used by a major U.S. packer. These schedules (schedules 1 and 2) are presented in table 2. Schedule 1 is currently in use by the packer, while schedule 2 is an older schedule.

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