theeﬃcacyofcontrollingphosphorusloading ... efficacy of controlling... ·...

The efficacy of controlling phosphorus loading:the case of management-intensive grazing§

Jonathan R. Winstena,*, Jeffrey R. Stokesb

aWinrock International State College, PA 16803-3475, USAbPenn State University, University Park, PA 16802, USA

Received 25 July 2002; received in revised form 6 May 2003; accepted 6 June 2003

Abstract

Consolidation in US agriculture has led to fewer, larger farms. In the case of dairy in theNortheastern US, higher concentrations of animals near large population centers pose water

quality problems that can be attributed to excessive soil nutrient levels. While new environ-mental policies and regulations are being developed and implemented to help manage suchproblems, research to determine the efficacy of alternative dairy production systems is needed.

The research reported in this paper makes use of stochastic dynamic programming to deter-mine optimal stocking densities, milk production levels, and feed rations for a hypotheticaldairy farm using management-intensive grazing. A key feature of the model is that financial

disincentives are placed on excessive accumulation of phosphorus in the farm’s soils. Theresults show that under optimal management the cost of reducing soil phosphorus to accep-table levels across all states of nature modeled is approximately $524 per hectare per year. Theoptimal farm management strategy is to rapidly reduce the size of the dairy herd (as opposed

to feeding for a lower level of milk production per cow) until soil phosphorus levels are undercontrol.# 2003 Elsevier Ltd. All rights reserved.

Keywords: Management-intensive grazing; Nutrient management; Phosphorus; Stochastic dynamic

programming; Dairy farming

0308-521X/03/$ - see front matter # 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/S0308-521X(03)00088-X

Agricultural Systems 79 (2004) 283–303

www.elsevier.com/locate/agsy

$ This research was partially funded by the Pennsylvania State University Agricultural Experiment

Station.

* Corresponding author. Tel.: +1-814-234-5666.

E-mail addresses: [email protected] (J.R. Winsten), [email protected] (J.R. Stokes).

http://www.sciencedirect.com



http://www.elsevier.com/locate/agsy/a4.3d

mailto:[email protected]

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1. Introduction

As with almost every other industry, dairy farming changed dramatically duringthe past century. Historical trends in the dairy industry reveal a decreasing numberof farms, increasing average herd size per farm, and increasing average milk pro-duction level per cow. The advancement of agricultural mechanization over thepast century allowed the average size of family-operated farms to increase signifi-cantly. Improvements in scientific understanding of agricultural systems has led tomanagement changes on farms and genetic improvement of crops and livestock.The trend toward larger herd sizes has been fueled by farmers seeking to capture

economies of scale which reduce average production costs per unit of output. Formany large farms, this strategy has proved successful resulting in increased profitsrelative to smaller farms. However, the financial investment required for large, con-finement facilities is prohibitive for many dairy producers. The managementrequirements of a confinement-feeding dairy farm include financial, biological,mechanical, and human resource skills. This, in conjunction with slim profit marginsin the dairy industry and extremely demanding operator labor requirements, hascontributed to the sharp decline in the number of dairy farms during the 1990s.In addition, more concentrated animal agriculture brought about by trends

toward larger dairy farms has, in many cases, had negative environmental con-sequences. Agriculture has been identified as the primary contributor to nonpointsource pollution (US Environmental Protection Agency, 1994). The problem stemsfrom excessive accumulation of nutrients in and on the soil, which can be trans-ported to surface and ground water through nutrient runoff, soil erosion, andleaching (Sharpley et al., 1999).The need to minimize nonpoint source pollution from agriculture has been the

subject of much policy dialogue during the 1990s. Currently, many states have reg-ulations on aspects of livestock farming in an attempt to reduce agricultural pollu-tion (e.g. CAFO regulations, bans on winter manure spreading, and minimummanure storage capacity). However, the regulations are often not well enforced andthere is no evidence that such policies are efficient or cost-effective for achievingreductions in agricultural pollution. Also, improper policy design has the potential toburden livestock farmers beyond what is necessary to meet policy targets.Previous research indicates that the use of management-intensive grazing (MIG)

for dairy production can result in increased farm profits and reduced phosphorusloading relative to conventional production practices (Winsten et al., 2000b). Thescientific principles underlying MIG were first described by the french agronomistAndre Voisin in the late 1950s (Murphy, 1994; Voisin, 1959). Management-intensivegrazing is a system in which cattle graze one section (paddock) of a larger pasturefor a short period of time, often only 12 or 24 h. The animals are rotated throughthe paddocks, allowing previously grazed paddocks to regrow (presumably to anoptimal level for nutrient yield) before regrazing occurs. For this reason, MIG isalso known as rotational, intensive-rotational, or short-duration grazing.Management-intensive grazing is a relatively new type of production system for

the US dairy industry. It is used by 5–20% of dairy farms in the conventional

284 J.R. Winsten, J.R. Stokes / Agricultural Systems 79 (2004) 283–303

dairybelt states and it represents an important future alternative for many conven-tional dairy farms (Winsten et al., 2000a). There has been limited research regardingoptimal dairy farm management under MIG. Preliminary research suggests thatthis system is economically competitive, while providing some positive environ-mental benefits relating to water quality (Winsten et al., 2000b; Owens and Bartho-lomew, 1995). This research contributes to the body of knowledge regarding MIGby modeling the dairy production system from the perspective of a profit max-imizing producer that potentially faces financial disincentives for high phosphorustest levels. The production system modeled has been pioneered by dairy farmers inNew Zealand: MIG coupled with a spring-calving schedule. The available dataindicate that the profitability of this system in the US can be well above average forthe industry and that the popularity of this system in on the rise (Winsten and Pet-rucci, 2002). However, it is paramount that we understand the environmentalimpacts of this system and balance those against farm profitability.The primary model is cast in a stochastic dynamic programing framework. Sec-

ondary models include linear and nonlinear programming models for balancingdairy feed rations and a Monte Carlo simulation model to assist in the summar-ization of the results.

2. Stochastic dynamic programming

The interactions between animal feeding, productivity, and nutrient excretion inconjunction with plant growth requirements and soil nutrient levels form a complexdairy farming system. By creating a dynamic optimization model that characterizesthis system, management insights can be gained which can potentially be used toreduce nonpoint source pollution. The approach taken is to internalize the costassociated with phosphorus accumulation on a representative Pennsylvania (PA)dairy farm by explicitly accounting for it in the farm’s profit function. The resultingoptimal management responses provide valuable information on minimum-coststrategies for reducing soil phosphorus accumulation and, hence, the potential forwater quality problems.The management responses examined in this research include changing herd size,

changing per cow milk production, and changing the farm’s feeding strategy. How-ever, the use of a multi-objective feeding strategy, which incorporates the goal ofreducing P excretion, resulted in identical feed rations as those from a standard cost-minimization feeding strategy for the milk production levels examined here.Although the multi-objective feeding strategy was shown to be a viable managementoption for higher average per cow milk production levels (10,920 kg per year), it willnot be discussed in this paper. Details on the multi-objective feeding strategy can befound in Winsten (2001).The problem facing the farm manager is to implement decisions that maximize the

sum of current and discounted future farm income under each state of nature ateach stage. Stages are defined by points in time when implementing decisions ispossible. Income is defined as the difference between gross farm receipts and all costs

J.R. Winsten, J.R. Stokes / Agricultural Systems 79 (2004) 283–303 285

of production and is dependent on the collection of state variables describing thesystem at each stage and a set of decision variables under managerial control. Grossfarm receipts arise from the sale of milk, calves, cull cows, and hay while variablecosts of production include feed, labor, and other variable costs, hay harvestingcosts, the cost of purchasing cows, and the cost (if any) associated with excess soiltest phosphorus (STP) levels.Let Vn be a function representing the expected net present value of the farms

income if an optimal policy is followed when there are n stages remaining in theplanning horizon and let Rn represent immediate income associated with an optimalpolicy. Both functions depend on five state variables while the immediate incomefunction additionally depends on two decision variables. The state variables are:average level of phosphorus in the farm’s soil, Pn, measured in kilograms (kg) STPper hectare (ha); herd size, HSn, measured as the number of cows on the dairy farm;hay inventory, HAYn, measured in metric tons of dry matter; per cow milkproduction, Mn, measured in average kilograms per day; and milk price, MPn,measured in dollars per 45.4 kg milk. The two decision variables are: how manycows to buy or sell, BSn, and whether or not to reduce milk production by feedingfor a lower lactation curve, LCn . Physiological constraints do not allow the animalto move to a higher lactation curve once the lactation has commenced.Dynamic programming has been used to address farm-level economic problems

since at least Burt and Allison (1963). Kennedy (1986) provides applications andbackground on dynamic programming, while a more recent compendium of dynamicprogramming applications to agricultural problems can be found in Taylor (1993).Decisions made for each state of nature at each stage influence the current stage’sincome and the likely resulting state of nature at the following stage. Bellman’srecursive relation (1957) can therefore be stated as

Vn Pn;HSn;HAYn;Mn;MPnð Þ ¼ maxBS;LC

�Rn BSn;LCnjPn;HSnHAYn;Mn;MPnð Þ

þ � � EmpVn�1;�Pn�1;HSn�1;HAYn�1;

Mn�1;MPn�1

��; ð1Þ

where b is the rate of time preference per stage, Emp is an expectation operator takenover the stochastic milk price state variable transition (described below), and max isthe maximization operator. The recursive relation highlights why a dynamic approachis useful in this setting. Current levels of soil P, the size of the dairy herd, hayinventory, milk production level, and milk prices are all information needed to makeoptimal stocking and milk production decisions. However, the levels for these statevariables and the ensuing optimal decisions have implications for the future state ofthe system (along with random price outcomes) and the recursive relation abovetakes this feature into account when the optimal decisions are determined. Thismodel is solved as a series of one-stage sub-problems where the last stage is the firstproblem solved using backward induction. Hence, backward numbering is evidentin Eq. (1).


2.1. State variables ranges

The levels of the five state variables determine the model’s state of nature at anygiven stage. These five variables have been selected as the minimum necessary tocharacterize the state of the system so that decisions can be made. However, not allstate variable values are accessible at every stage as described below.The herd size range is from 50 to 200 cows with an interval of 10 cows. This range

was selected because the stocking rates represented (0.1–0.4 cows per hectare) shouldencompass the level that is likely to result in a long-term soil phosphorus balance onthe farm, given the amount and estimated productivity of the land. An interval of10 cows was selected because it is large enough to impact profitability and thesoil phosphorus level, but is not unduly burdensome for model execution and resultspresentation.The milk production state variable represents kilograms of milk produced per cow

per day. This variable assumes the value of 0 during the dry months and ranges from10 to 36 kg per cow per day over the lactation. The lactation curves suggested bythis can be seen in Fig. 1. The values are arranged in each month to form three dis-tinct lactation curves which represent low, moderate, and high milk productionlevels per cow over the lactation. The low milk production level represents 5429 kg/cow/year (approximately 12,000 lbs). The moderate production level represents 7260kg/cow/year (approximately 16,000 lbs). The high production represents 9090 kg/cow/year (approximately 20,000 lbs).This milk production range is lower than that found on conventional and con-

finement feeding farms. Milk production per cow is constrained on farms using theproduction system being modeled in this research. Management-intensive grazing

Fig. 1. Lactation curves included in the model. LC represents the binary decision to remain on the current

curve or transition to a lower lactation curve at the next stage.


can limit milk production potential due to energy expenditure by the cows anddifficulty in balancing rations. Spring calving can limit per cow milk productionbecause the lactation peak comes during the heat of the summer and because theaverage lactation length will be less than 305 days to maintain the seasonal schedule.Because the herd being modeled is on a spring calving schedule, the entire herd is

assumed to be dry for the months of January and February. Milk productioncommences in March (upon calving) and continues through December. During thelactation, the model selects the most profitable, in a dynamic context, of the threelactation curves available. With this modeling convention, only one milk produc-tion level is evaluated for the dry months and three for each month of lactation.The range of the P state variable is from 36 to 125 kgs/ha in 0.9-kg increments.

The increment needs to be small to capture the small monthly changes that canresult from the decisions modeled. Although STP levels are not a perfect predictorof P runoff or water quality problems, they are correlated with P runoff (Sharpley etal., 1999). Numerous soil characteristics and land management practices will affectthis relationship. Phosphorus in the soil is necessary for plant growth. However,the analysis of soil samples in many livestock producing states are revealing thatmore than 50% of samples contain P at levels beyond the need for any P fertilization(Fixen and Grove, 1990). Many of these high P soils exist in the watershed ofP-sensitive waters, such as the Great Lakes, Chesapeake Bay, and the Everglades.The optimum range for STP levels is often defined as 25–50 ppm (using the Mehlich-1extractant), or roughly 45–89 kgs P/ha, and greater than 50 ppm is defined asexcessive (Fixen and Grove, 1990). Therefore, STP levels above 89 kg/ha are con-sidered excessive as a general rule in this model.The milk prices modeled range from $9.50 to $17.50 per 45.4 kg milk. This range

covers the vast majority of monthly milk prices as measured by the basic formulaprice (BFP) during the 1990s. The increment for the milk price state variable is $1 per45.4 kg milk. This state variable is the only one in the model that transitions sto-chastically. This is described below in the section on state variable transition equations.The hay stock state variable range is from 0 to 362 metric tons (t), with an incre-

ment of 91 t. This variable represents the amount of hay that is accumulated instorage on the farm at any given stage. While 91 t is a large increment, the hay stockstate variable does not have a large influence on the model’s objective function.Therefore, efficient modeling dictates that such state variables assume a limitednumber of values as suggested by Burt (1982).The total number of encountered states of nature, also known as the dimensions

of the model’s state space, is the product of all accessible state variable values ateach stage. This is 110,160 for each month of lactation (March through December)and 36,720 for each dry month (January and February). The annual state space forthis model then represents 1,175,040 states of nature.

2.2. Decision variables

The decision to buy or sell cows, BSn, represents a change in the size of the herdthrough the purchase or culling of cows from the existing herd. This is an important


management decision that impacts pasture utilization, phosphorus excretion, andfarm profit. It is assumed that reductions in the herd size will be optimal whendisincentives are placed on accumulating STP levels.The values that the BSn decision variable can take range from �30 to +20 by 10

cow increments, where �30 represents buying thirty cows and +20 represents selling20 cows. Because of the seasonal calving schedule the model can buy cows only inMarch at freshening to ensure simultaneous lactation cycles of the entire herd.Therefore, for every state of nature encountered in March the decision alternativesare to buy 30, 20, or 10 cows or to sell 0, 10, or 20 cows. In all other months, thedecision alternatives are to sell 0, 10, or 20 cows only. The decision alternatives aremodified at the endpoints of the herd size state variable to ensure that the resultingherd size remains in the range being modeled.The lactation curve decision variable, LCn, represents the decision to reduce milk

production levels to the next lower lactation curve for the next stage. Because themilk production level per cow also impacts profitability and STP levels, this decisionprovides the model with a management response to penalties on accumulating soilphosphorus. Lower milk production levels will result in lower phosphorus excretion,ceteris paribus. However, this will also reduce per cow profitability. The trade-offbetween milk production and accumulating soil phosphorus is not as clear, a priori,as that between additional cows and accumulating soil phosphorus.The lactation curve decision is modeled as a binary choice variable. It is executed

in the state transition equation for the milk production state variable. This decisionis analyzed only when the stage and state of nature make it possible to produce onthe next lower lactation curve at the next stage. For example, when the month isDecember or January or if the state of nature is on the lowest lactation curve thelactation curve decision is irrelevant (see Fig. 1). The physiological limits of thedairy cow suggest that it is not possible to move to a higher lactation curve once thelactation has commenced.

2.3. State variable transition equations

The state transition equations are the intertemporal link between the currentstage’s state of nature and the ensuing state of nature at the following stage, basedon decisions made at the current stage. Four of the state variables transitionaccording to deterministic relationships, while the milk price transition is stochastic.The soil phosphorus level at any subsequent stage, Pn�1, is equal to the current

level, Pn, plus additional phosphorus that is contained in the excreted manure (pmn)less phosphorus that is absorbed for pasture forage growth (fn). The function fn isderived from a set of representative monthly pasture growth parameters multipliedby the average phosphorus content of the forage in that month. The transitionequation is

Pn�1 ¼ Pn þ pmn � fnpmn ¼ g HSn;Mn;FRnð Þ ð2Þ


FRn ¼ h Mn; �ð Þ:

Manure P, pmn, is shown as a function, g(.), of the size of the herd, the milk pro-duction level, and the current stage’s feed ration, FR n, which is itself a function, h(.),of the milk production level and a vector of exogenous parameters denoted by y.The nature of the g(.) and h(.) functions are discussed below in detail. Because thesevalues will not map exactly into those future state variable values modeled, linearinterpolation is used on the value function for the P state variable.The herd size state transition equation is simply where next stage’s herd size is

equal to the current stage’s herd size less cows bought or sold based on the decisionvariable BSn. Therefore, the equation is

HSn�1 ¼ HSn � BSn: ð3Þ

The hay stock transition equation is a simple accounting relationship which equatesthe amount of hay inventory at the next stage (HAYn-1) to that in the current stageplus hay produced (hmn) plus hay purchased (hbn) less hay sold (hln) less hay fed (hfn)in the current period. This relationship is represented by the following equation

HAYn�1 ¼ HAYn þ hmn þ hbn � hln � hfn: ð4Þ

The amount of hay produced (hmn) is the accumulated forage on land that is notgrazed in that stage less a constant harvesting loss parameter. Hay fed (hfn) in thecurrent period is determined by the herd size and the optimal ration formulation (seeoptimal feed rations section below). Hay purchases (hbn) happen when the hay stockis less than amount of hay fed in the optimal ration. Whenever the stage is equal toMay and the current hay stock is greater than zero, the existing hay stock is sold(hln) to make room for the coming season’s hay production.The milk production state variable transition can be understood most easily while

viewing Fig. 1. For example, if the current month is June and the milk productionstate is 40 kg/cow/day, then the lactation curve decision determines the most prof-itable milk production level for July (either LCn=0 and 38 kg/cow/day or LCn=1and 32 kg/cow/day). This transition equation is limited to conform to the physi-ological limits of the cow’s lactation curve. Therefore, it is not possible to moveto higher curves during the lactation. An example milk production state variabletransition is

Mn�1 ¼ 1� LCnð Þ � Mn � 2ð Þ þ LCnð Þ � Mn � 8ð Þ: ð5Þ

When the decision is to remain on the current lactation curve (LCn=0), the milkproduction level at the subsequent stage (Mn-1) is reduced by 2 kg/cow/day. Whenthe decision is to shift to the next lower lactation curve (LCn=1), the milk produc-tion level at the subsequent stage (Mn-1) is reduced by 8 kg/cow/day.The transition equation above represents an example of a milk production tran-

sition equation that could be applicable from May to November. In early lactation,this equation is modified to reflect the increasing portion of the lactation curve.Because milk production is zero during the dry months, the lactation curve decisionis irrelevant in December and January.


Milk prices are exogenous to this system and evolve stochastically. Using the basicformula price (BFP) as the relevant price dairy producers use to make productiondecisions, econometric tests reveal no evidence of seasonality. Monthly BFP datawere collected from January 1990 through June 2001 from the National AgriculturalStatistical Service. Eleven monthly dummy variables were modeled and jointly testedfor significance. The F-test statistic was calculated at 1.72. With 11 and 112 degreesof freedom, the null hypothesis of a seasonal component in the BFP is rejected at the5% level. A first order autoregressive [AR(1)] process appears to adequately capturethe movement in the BFP. The milk price state variable transition equation wasestimated as

MPn�1 ¼ 2:313:55ð Þ

þ 0:802713:11ð Þ

MPn þ "~n where "~n � N 0; �ð Þ; ð6Þ

where t-statistics are reported in parentheses.The errors generated from this model were used to approximate conditionally

normal cumulative density functions (CDFs) for one-period stochastic milk pricetransitions according to Taylor’s hyperbolic tangent transformation (Taylor, 1984).Using Taylor’s method, the estimated CDFs are

F "ð Þ ¼1

2þ1

2tanh �0:04255

�0:56ð Þþ 1:14369"

13:11ð Þ

� �; ð7Þ

where E represents a vector of errors associated with the AR(1) process from Eq. (6),the bracketed expression represents the best relationship estimated using Taylor’smethod based on the Schwarz model selection criterion (t-statistics are shown inparentheses), and F(E) represents the cumulative probability of one-stage milk pricetransitions.The resulting matrix of transition probabilities are shown in Table 1. Each row

represents a milk price at stage n and each column represents a milk price at stagen�1. The cells of the table, therefore, show the probability associated with movingfrom any current milk price to any future milk price over one stage (i.e. in con-secutive months). The milk price transition probabilities were calculated by usingeach row and column combination of milk prices from Table 1 and determining theerror using Eq. (6). The calculated error was then used in Eq. (7) to determine therelevant probability.

3. Production assumptions and data

While the type of farming system being modeled in this research is not widelyimplemented in the US, an example is Cove Mountain Farm (CMF) in FranklinCounty, PA. Detailed cost information on the establishment of CMF is presented inWinsten and Petrucci (1999). The farm being modeled is assumed to have 81hectares of open land, all of which is maintained as permanent pasture land. Thefarm infrastructure includes a 16-unit New Zealand-style ‘‘swing’’ milking parlor.


The details related to this production system as well as information on all of the dataused in this research can be found in Winsten (2001).Perhaps the most important parameter in this model represents the cost of accu-

mulating STP levels. This cost parameter is varied to create different scenarios of themodel and is assessed as dollars per kg of STP per hectare at STP levels above 89kgs per hectare, above which STP levels are generally considered excessive (Fixenand Grove, 1990). It is important to understand that it is not assumed nor suggestedthat this phosphorus loading ‘‘penalty’’ is a policy tool that will be implemented inthe future. This research is attempting to discover optimal producer behavior whendisincentives are in place to minimize accumulating STP levels. Toward this end, thisparameter could have been equally modeled as a subsidy to the producer for reducedSTP levels.

4. Optimal feed rations

A separate, mathematical programming model is used to determine the optimalfeed ration for the herd based on the stage and state of nature of the SDP model.Although the ration balancing (RB) model is a separate entity from the SDP model,it is state dependent and has important implications for future P levels in the SDPmodel. The RB model is solved at each stage for each state of nature in the SDPmodel using linear programming methods to determine the optimal feed rations..The SDP model provides information regarding the state of the system (i.e. thestage, herd size, and per cow milk production level) to the RB model for use inconstructing relevant constraints. The RB model passes the necessary informationabout the optimal feed ration (i.e. amount of pasture forage, hay and other ingre-dients, ration cost, and estimated amount of resulting P excretion) back to the SDPmodel.

Table 1

Milk price transition probabilitiesa

Milk Price at stage n
Milk price at stage n�1
$/45.4 kg
$9.50 $10.50 $11.50 $12.50 $13.50 $14.50 $15.50 $16.50 $17.50
$9.50
0.7439 0.2223 0.0302 0.0032 0.0003 0.0000 0.0000 0.0000 0.0000
$10.50
0.3165 0.5037 0.1580 0.0195 0.0020 0.0002 0.0000 0.0000 0.0000
$11.50
0.0688 0.3523 0.4565 0.1085 0.0125 0.0013 0.0001 0.0000 0.0000
$12.50
0.0116 0.0923 0.4292 0.3852 0.0727 0.0080 0.0008 0.0001 0.0000
$13.50
0.0019 0.0163 0.1359 0.4880 0.3044 0.0479 0.0051 0.0005 0.0001
$14.50
0.0003 0.0026 0.0253 0.1941 0.5156 0.2272 0.0312 0.0033 0.0004
$15.50
0.0000 0.0004 0.0041 0.0390 0.2663 0.5057 0.1620 0.0201 0.0023
$16.50
0.0000 0.0001 0.0007 0.0065 0.0596 0.3467 0.4606 0.1115 0.0144
$17.50
0.0000 0.0000 0.0001 0.0010 0.0101 0.0898 0.4244 0.3906 0.0840
a The numbers in the table represent the probability of moving from any price at stage n to any other

price at the next stage (n�1). They were estimated using a procedure developed by Taylor (1984).


Cost-minimization ration balancing has often been criticized as failing to discovertruly profit maximizing feed rations, because the output level is held constant.However, because the SDP model analyzes several milk production levels in eachmonth, it can be argued that the feed rations selected are closer to profit maximizinglevels than would otherwise be the case.Cost minimization rations are determined in the conventional manner (see, for

example, Church, 1991). Six nutrients are modeled: crude protein, undegradedintake protein, net energy of lactation, neutral detergent fiber (NDF), calcium, andphosphorus. Data on nutrient requirements used in the RB model are from the 1989National Research Council (NRC) recommendations. However, P requirementshave been adjusted to reflect the numerous recent research findings that show dairyrations only require from 0.3 to 0.4% of ration dry matter as phosphorus, asopposed to 0.48% (Wu and Satter, 2000; Morse et al., 1992). As indicated above,nutrient requirement limits used in the RB model are a function of the per cow milkproduction level, Mn, dictated by the current state of nature of the SDP. Four con-straints are also included in the RB to insure dry matter intake per cow (DMI), totalherd pasture consumption, pasture consumption per cow, and NDF consumptiondo not exceed physical limitations. Total herd pasture consumption is calculated asthe product of pasture fed per cow and the herd size state variable, HSn. Pastureconsumption per cow must be less than or equal to 3% of the body weight, or 15 kgDM/day.In addition to pasture forage and conserved hay, 12 feeds are available to the

ration balancing model. These feeds were chosen as the most appropriate and widelyused supplements for dairy cattle on pasture. The nutrient composition of thesefeeds was also taken from the 1989 Nutrient Requirements of Dairy Cattle (NRC,1988) while prices for these feeds are representative of their market values in the year2001. The nutrient composition of the pasture forage was taken from data collectedand published by Muller and Fales (1998). This data represents mixed, mostly grasspasture for each month through the growing season.In addition to the amount of each feed ingredient used in each ration, the output

from the RB model specifies the ration cost and estimated phosphorus excretion percow, which is used in the SDP model to determine the STP level, current profits andthe likely future state of the system. The total phosphorus excretion by the herd fora given month, pmn, is calculated in the SDP model. It equals phosphorus excretionper cow per day multiplied by the herd size state varaible, HSn, and the days permonth.

5. Simulation procedure

The convergent optimal decision rules for the SDP discussed above containinformation for almost 1.2 million unique states of nature over a 12-month period.For a problem of this size, it is not practical to present the entirety of this informa-tion. For this reason, a new approach for summarizing SDP results was developed.Mjelde et al. (1992) discuss the application of Markov process theory to summarize


DP models with multiple transition matrices. Their approach could be used in thepresent research as there are 12 transistion probability matrices in the SDP model.However, an alternative approach of using Monte Carlo simulation is undertakenhere. While also capable of summarizing a large state-space, such an approach hasthe advantage of allowing for the determination of how much time it takes for thesystem to gravitate toward acceptable STP levels given pre-determined initial con-ditions. Estimating this amount of time is an important and policy-relevant issuethat has been identified in the P literature (Sharpley et al., 1999).The approach uses the optimal decision rules generated by the SDP model in a

Monte Carlo simulation. Because every state of nature is accessible (eventually)from every other state of nature, the optimal decision rules lead to the model’sconvergent state values, regardless of the initial state of nature. Therefore, once theoptimal decision rules are determined from the SDP model, they are used in aseparate program to construct optimal paths for the state variables to the con-vergent state. While this approach for summarizing the results of SDP models mayhave been used previously, examples are not evident in the literature.It should be understood that the simulation procedure does not determine the

optimal decisions, but rather applies the optimal decisions determined by the SDPmodel to reveal optimal paths of the state variables through time. The simulationitself is straightforward. A randomly chosen state of nature is initially determinedand the relevant optimal policy is applied to determine, via the state transitionequations, which state of nature will prevail during the next stage. This procedure isthen repeated until the convergent state values (i.e. the long-run value of each statevariable) are determined.Two sources of randomness make this simulation approach stochastic. First,

consistent with monte Carlo simulation, only a subset of sample paths are usedbecause the state space of the SDP is large. Therefore, the initial state of the systemis determined at random. Depending on the scenario under investigation (e.g. a farmsituation characterized by very high STP or very low STP), starting conditions canbe sampled from a restricted set of state variable values. Second, the milk price statevariable transition is stochastic implying that a random error must be drawn todetermine the next milk price and hence state of nature.

6. Model results and discussion

As noted above, the simulation procedure was designed to characterize the systemgiven the size of the state space. The model’s convergent state for a particular sce-nario is determined by applying the optimal decision rules over a number of stagesand is designed to determine which states of nature the system gravitates toward inthe long-run. The convergent state analysis for each scenario starts from 10,000randomly chosen initial conditions (i.e. states of nature) and simulates an optimalpath through time. The convergent state indicates the state variable values, farmprofit level, and the percentage of nutrients imported onto the farm that result fromapplying the optimal decision rules, regardless of the initial state of nature.


Of particular interest in this research are changes in the convergent state resultingfrom changes in the phosphorus penalty parameter. The simulation procedure thatproduces the convergent state results is shown to be an effective means for sum-marizing the results (i.e. the optimal decisions) from this large SDP model and, initself, makes a contribution in the area of methodologies for summarizing large SDPmodels.

6.1. Baseline scenario

The distinguishing characteristic of the baseline scenario is that no disincentivesexist to control phosphorus loading on the farm (i.e. the phosphorus penalty iszero). The convergent state for the baseline scenario is, therefore, characterized bythe maximum herd size (200 cows), maximum per cow milk production (9090 kg),and maximum STP levels (125 kg/ha) irrespective of the initial state of nature. Theseresults can be seen in Table 2. Regardless of the initial herd size and STP level, theoptimal decisions lead the profit-maximizing producer to this convergent state levelwithout exception. This result is also consistent with economic theory, which statesthat output should be increased as long as the marginal revenue (MR) of additionaloutput is greater than the marginal cost (MC) of producing the output. The optimaldecisions show that MR is greater than the MC for the high milk production levelregardless of the milk price. Therefore, the convergent state gravitates to the highestlactation curve being modeled as the optimal level.Average annual profits for the baseline scenario are $66,643 or $333 per cow. To

achieve these results, this scenario requires that 62% of the herd’s nutrient intake be

Table 2

Convergent state results under the baseline scenarioa

Herd Size

cows

Milkb

kg/cow/day

Hay

tons

STP

kg/ha

Milk

Price $/cwt

Profit

$

Imported

nutrientsc (%)

January
200 0 0 123 12.07 �29,641 100.00
February
200 0 0 125 12.07 �29,641 100.00
March
200 32 0 123 12.06 11,326 100.00
April
200 34 0 123 12.05 20,354 55.76
May
200 36 0 125 12.06 19,687 38.24
June
200 34 91 123 12.06 14,136 53.37
July
200 32 181 123 12.06 16,221 49.81
August
200 30 181 123 12.07 14,590 48.57
September
200 28 181 123 12.08 9468 51.93
October
200 26 181 123 12.08 10,085 50.32
November
200 24 181 125 12.08 6218 49.58
December
200 22 91 123 12.09 3839 45.61
Total/average
200 9089 91 124 $12.07 $66,643 61.93
a The numbers in this table are averages of the states of nature resulting from applying the optimal

decisions from SDP model from 10,000 randomly chosen initial states of nature when no disincentive for

phosphorus loading exists. The variance of the herd size and milk production results are zero.b The total milk production represents kg/cow/year.


satisfied from feeds that are imported onto the farm. These imported nutrientsinclude purchased hay, corn silage, and grains. The stocking rate in the convergentstate is 2.47 cows/ha (the maximum allowed in the model), which is substantiallygreater than that of most pasture-based dairy farms in Pennsylvania. The high con-vergent state stocking rate results in an annual forage requirement for the herd thatis greater than the pasture growth can supply. Because of this, hay inventory is 0 tfrom January through May. This result implies all fed hay fed in these months mustbe purchased.Although we do not often see a level of stocking density as high as 2.47 cows/ha

on pasture-based dairy farms in Pennsylvania, the use of highly labor-efficientmilking parlors and out-wintering could make this result increasingly frequent.These results show that the profit-maximizing stocking density when disincentivesfor phosphorus loading do not exist will lead to increasing STP levels over time.Such a situation represents a potential threat to water quality, particularly with closeproximity to surface water and/or with uneven distribution of manure nutrients onthe farm.

6.2. Phosphorus control scenario ($13 penalty)

The financial penalty for STP levels in excess of 89 kg/ha was incrementallyincreased in the model until P loading was reduced under every state of nature. Thissection presents the convergent state that results from the existence of a $13.38/kgSTP/ha financial disincentive for phosphorus loading. The convergent state infor-mation presented in Table 3 results from optimal decisions (from the SDP model)that are based on this level of disincentive. The actual cost of the phosphoruspenalty is not explicitly accounted for in the simulation model that produced theconvergent state results. This has two important advantages. First, it reduces thedependence of the convergent state results on the design of the phosphorus loadingpenalty. Second, the cost of reducing phosphorus loading through optimal manage-ment actions can be viewed as an opportunity cost based on foregone farm profit. Inthis way, a range of possible policy actions that induce optimal management decisionscan be considered as alternatives to a penalty on excess phosphorus accumulation.As shown in Table 3, the average herd size throughout the year is 103 cows.

Slightly greater average herd sizes are indicated during the grazing months of Aprilthrough October. Fig. 2 shows the frequency distribution by month of the resultingconvergent state herd sizes from 10,000 simulated, optimal paths that originate fromrandomly selected states of nature. The slightly greater average herd sizes during thegrazing months is due to at least two factors. First, the increased phosphorus uptakeresulting from pasture growth offsets some of the phosphorus from having addi-tional cows. Second, per cow milk production is higher in these months whichincreases the demand for additional cows to maximize profitability. The resultingherd sizes appear to be distributed symmetrically around the respective monthlymeans. The distribution of convergent state herd sizes in this scenario is caused bythe wide variety of initial states of nature, as well as the stochastic nature of milkprices along each path.


Referring back to Table 3, the convergent state milk production level is the max-imum allowed in the model, with an annual yield of 9090 kg/cow. There was novariation in this result; that is, all 10,000 simulated paths settle on the highest lac-tation curve in the convergent state. This result is not wholly unexpected in this casebecause milk output per unit of phosphorus intake increases as the percentagerequired for animal maintenance decreases at higher milk production levels. Thisimplies that higher milk production levels will be optimal in the face of phosphorusloading disincentives. However, it should be stressed that this result is likely depen-dent on the feed prices used in the model. Additionally, while not modeled explicitly,it should be noted that higher milk production levels often result in increased phos-phorus utilization efficiency by the cow (Morse et al., 1992). This result should bereflected in the NRC’s nutrient requirement recommendations for phosphoruswhich are used in the RB model.Hay stocks remain close to their maximum value of 363 t throughout the year in

the convergent state. Table 3 also shows that hay stocks are at 363 t every month.This is partially due to rounding the hay stock state variable to the nearest 91 tincrement in the simulation model. Hay fed in the winter months is between 23and 27 t per month. Some corn silage is fed as an energy source to the dry cows inJanuary and February.Examples of the feed rations used in each month can be seen in Table 4. The cost

of the feed ration is lowest ($0.73 per cow per day) in January and February whenthe herd is dry. The cost increases greatly to $2.66 in March when milk production is32 kg/cow/day and pasture forage is unavailable. The primary ration ingredientsare hay and ground corn. This results in phosphorus excretion of 47 g/cow/day.From April through October, available pasture is the only forage source for theherd. Phosphorus excretion is at its highest level in May (51 g/cow/day) and thendecreases as does the milk production level.

Fig. 2. Average annual frequency distribution of herd size under the P control scenario.


This scenario results in an annual average of 45% of feed nutrients imported ontothe farm. It is very important for environmental quality that nutrients be carefullymanaged. The feeding of imported corn silage is reflected in the percentage ofnutrients imported (last column in Table 3). During the non-lactating months, Jan-uary and February, the herd’s nutrient requirements are met primarily with hay andcorn silage. The percentage of nutrients imported during these months are the lowestof the year. The percentage of nutrients imported during the growing season, Aprilthrough October, reflect the grain supplementation needed to meet milk productionrequirements. The forage requirement during these months is generally satisfiedfrom pasture.The model allows the average convergent state STP levels to vary throughout the

year from 63 to 70 kg/ha as shown in Table 3. There is a clear trend in the STP levelthroughout the year; reaching a high in April and May, decreasing throughout thegrazing season, and accumulating in the winter months. There are two importantfactors influencing this trend. First, during the grazing season, pasture is supplyingalmost all of the forage nutrients to the herd. Even though the herd is being sup-plemented daily an average of 8.3 kg/grain/cow, phosphorus excretion from 103cows tends to be less than phosphorus uptake by the pasture growth. The decreasedrate of STP reduction from August to September reflects the decreased pasturegrowth rate in August (the summer slump). Second, decreases in the herd’s milkproduction level after May reduces the phosphorus intake requirement and is likelyto reduce total phosphorus excretion, even though phosphorus utilization efficiencyis reduced at lower milk production levels. This trend contributes to the reduction in

Table 3

Convergent state results under the phosphorus control scenarioa

Herd Size

cows

Milkb

kg/cow/day

Hay

tons

STP

kgs/ha

Milk

Price $/45 kgs

Profit

$

Imported

nutrientsc (%)

January
103 0 363 66 12.07 �15,048 33.03
February
103 0 363 67 12.07 �15,065 33.03
March
103 32 363 68 12.06 1994 60.00
April
104 34 363 70 12.05 8328 34.90
May
104 36 363 70 12.06 38,316 38.24
June
104 34 363 68 12.06 �634 53.37
July
104 32 363 66 12.06 2422 49.81
August
104 30 363 64 12.07 2155 48.57
September
104 28 363 64 12.08 �1104 51.93
October
104 26 363 63 12.08 1695 48.55
November
103 24 363 63 12.08 1198 49.58
December
103 22 363 64 12.09 �55 45.61
Total/Average
103 9089 363 66 $12.07 $24,201 45.55
a The numbers in this table are averages of the states of nature resulting from applying the optimal

decisions from the SDP model from 10,000 randomly chosen initial states of nature when a penalty of

$15/lb on excess phosphorus exists. The variance of the milk production and hay inventory results are

zero.b The total milk production represents kg/cow/year.


STP levels. The STP levels clearly accumulate during the non-grazing months whenexcreted phosphorus is not balanced by uptake from pasture growth.With the average of the convergent state milk prices at $12.07, the convergent

state profitability under this scenario averages $24,201 per year, or $235 per cow. Asin most seasonal dairy operations, the dry months have negative cash flow. This canbe seen in the results for January and February in the top two rows of Table 3. Theprofitability in March is enhanced by the sale of all calves as is the profitability inMay from the sale of surplus hay. During the grazing months, the value of themarginal product of feed is much higher than in non-grazing months. This is due tothe use of relatively inexpensive pasture forage to meet the herd’s requirements andcan be seen in the last column of Table 4. Total profits in the grazing months arereduced by the cost of mechanically harvesting excess pasture forage as hay.Relative to the baseline scenario, which results in a stocking rate of 2.47 cows/ha,

the P control scenario will cost the farmer $42,442 in foregone profit on average.The optimal management strategy for controlling P loading calls for a stocking rateof roughly 1.24 cows/ha. However, most pasture-based dairy farms in the US havestocking rates less than 1.24 cows/ha. A 1996 survey by Winsten et al. (2000a) repor-ted a stocking rate of 0.84 cows/ha on Pennsylvania dairy farms using MIG. Thetrend on seasonal calving, pasture-based farms seems to be toward greaterstocking rates to more fully utilize pasture forage and increase profits. Increasingstocking rates are being implemented at the expense of average per cow milk pro-duction levels (see Winsten and Petrucci, 2002). This research indicates that a moreprofitable strategy is to keep the stocking rate near 1.24 cows/ha and manage forhigher per cow milk production levels when a disincentive exists for excess Paccumulation.

6.3. Sensitivity to pasture growth rate

The results presented in the previous sections are based on average pasture growthrates for mixed, mostly grass pastures in Pennsylvania. Since pasture growth is theonly acceptable phosphorus ‘‘sink’’ in this model, this section examines the impactof changes in the pasture growth rate on the convergent state results. The baselineand phosphorus control scenarios were modeled using average pasture growth rates.The annual forage dry matter (DM) production for the average growth rate is 7.9 tDM/ha. This section presents convergent state results for the phosphorus controlscenario when the annual pasture growth is low (5.45 t DM/ha) and when theannual pasture growth is high (13.6 t DM/ha).When pasture growth is low, the convergent state herd size is reduced to 73 cows

on average throughout the year. The convergent state milk production level remainson the highest lactation curve for the same reasons discussed previously. The level ofhay stocks are reduced by approximately 44 t compared with the average pasturegrowth rates. This reduction is ameliorated by the lower average herd, thereby con-suming less hay. The convergent state STP levels are reduced from 66 to 60 kgs/ha. Theaverage profits earned by the farm when pasture growth is low is reduced to $15,096.This is $9105 dollars less than that earned with average pasture growth.


3

Table 4

Examples of convergent stage minimum cost rations by montha (kg/cow/day unless otherwise stated)

Ration cost

($/cow/day)

Phosphorus

excretion

(g/cow/day)

Pasture

(kg)

Hay Corn

silage

Soybean

meal

Canola

meal

Cotton

seed

Distiller’s

grain

Ground

corn

High

moisture

shell corn

Ground

barley

Milk

productionb

(kg)

Milk

income

per gram

P excretionc

Milk

income

over

feed costsd

January $0.73 28.40 0.00 5.80 2.84 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0 $0.00 �$0.73

February $0.73 28.40 0.00 5.80 2.84 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0 $0.00 �$0.73

March $2.66 47.10 0.00 7.72 0.00 1.14 1.01 0.00 0.00 9.27 0.00 0.00 32 $0.18 $5.86

April $1.83 49.10 12.26 0.00 0.00 0.83 0.00 0.00 0.00 5.60 0.00 0.00 34 $0.18 $7.22

May $1.98 51.00 12.05 0.00 0.00 0.95 0.00 0.00 0.00 6.38 0.00 0.00 36 $0.19 $7.60

June $2.07 49.10 9.05 0.00 0.00 0.00 1.29 0.00 0.00 9.00 0.00 0.00 34 $0.18 $6.98

July $1.92 47.10 9.40 0.00 0.00 0.00 0.98 0.00 0.00 8.27 0.00 0.00 32 $0.18 $6.60

August $1.85 45.10 9.16 0.00 0.00 0.00 2.16 0.00 0.00 6.47 0.00 0.00 30 $0.18 $6.13

September $1.84 43.20 7.96 0.00 0.00 0.00 1.39 0.00 2.59 4.61 0.00 0.00 28 $0.17 $5.61

October $1.71 41.20 8.15 0.00 0.00 0.00 1.09 0.00 2.82 3.78 0.00 0.00 26 $0.17 $5.21

November $1.95 39.30 0.00 8.42 0.00 0.00 0.98 0.00 0.00 7.19 0.00 0.00 24 $0.16 $4.44

December $1.81 37.30 0.00 8.73 0.00 0.00 0.65 0.00 0.00 6.56 0.00 0.00 22 $0.16 $4.05

a This table contains a subset of results from the ration balancing model used to determine the minimum cost feed ration.b The rations shown are based on milk production levels from the highest lactation curve.c Represents the daily milk revenue (based on the mean price of $12.07/cwt.) divided by the daily grams of P excretion.

00

J.R.Winsten

,J.R.Stokes/Agricu

lturalSystem

s79(2004)283–303

When pasture growth is high the average herd size in the convergent state is 176cows. Like all the other scenarios, milk production remains on the highest lactationcurve without exception. The herd size and milk production level result in an averageannual farm profit of $53,678; more than twice that resulting from average pasturegrowth. The convergent state STP level is lower under high pasture growth thanunder average or low growth conditions, while the average growth rate resulted inthe highest average STP levels. The relationship between herd size and pasturegrowth rate must determine the resulting STP levels, but the exact relationship is notclear.The sensitivity of the convergent state values to changes in the pasture growth

rates provides interesting information. The rate of pasture growth determines howmuch phosphorus will be taken up from the soil. This has a large influence on con-vergent state herd size and resulting farm profit. However, due to the stochasticnature of weather, accurate management decisions cannot be made that reflectfuture pasture growth conditions. If a farm manager were to implement decisionsbased on average growth conditions, these results imply that excess phosphoruswould result if low pasture growth occurs and that foregone profits will represent anopportunity cost if high pasture growth occurs.

6.4. Expected time to reach acceptable STP levels

In addition to using the optimal decision rules to determine the convergent stateresults from random initial conditions, they were also used to measure the numberof years required to reach an acceptable STP level from specific starting states ofnature. This was done with slight modifications to the simulation procedure descri-bed above. This analysis is motivated by the need to know how long it would takefor a farm to reduce STP from high to acceptable levels while acting according to theoptimal economic behavior. If the time required to affect such change is beyond theplanning horizon of policy makers, then policies that invoke these managementdecisions may be deemed ineffective and may need to be revised.This analysis measured the number of years for the farm’s STP level to be reduced

to 71 kg/ha from states of nature that would be considered ‘‘problematic’’ from aphosphorus loading perspective. The initial states of nature are those where the farmhas high STP levels (from 116 to 125 kg P/ha) and a large herd size (from 180 to 200cows). Ten-thousand paths are traced using the simulation model from initial statesof nature drawn randomly from these limited ranges, with the other state variablesheld at moderate values. The stages are counted along each path and the stage atwhich the STP level first reaches 71 kg/ha is noted.The time required to reduce the STP level to 71 kg/ha was estimated to be just

under 5 years on average. While 94% of the simulated paths were able to reach 71kg/ha in 6 years or less, some paths required as long as 17 years. The primaryinfluence on the rate of STP reductions is the herd size along the optimal path. Asexpected, the paths that reach the target STP level the fastest are those that reducecow numbers the fastest. The driving force influencing the rate of change in the herdsize is the price of milk, which has a significant impact on the optimal buy/sell


decisions. Simulated paths where the stochastic movement of milk prices reflectsunusually high levels will induce higher herd sizes along the path, thus slowing therate of STP reductions.This discussion highlights a potential conflict between farm support policies and

environmental policies. Any policy designed to increase the price of milk will likelyincrease the demand for cows. This could increase the stocking rate on dairy farmsand exacerbate phosphorus loading problems.Policymakers need to be aware of the inadvertent consequences of farm support

policies that have the potential to conflict with other policy priorities. The formerNortheast Dairy Compact presents a good example of this. The Compact, whichincreased the milk price received by farmers who sell milk in the participatingstates, could increase the phosphorus loading problem in the Lake Champlain, aphosphorus-sensitive water body.

7. Conclusions

Consistent with the real world, the results show that with no economic incentive toconsider the issue of accumulating soil phosphorus, the model chooses decisions thatlead to increased herd sizes and maximum per cow milk production levels. Thisresult is also consistent with economic theory and matches a priori expectations.Given the profitability of this farming system, there is a substantial opportunity cost($524/ha/year) associated with maintaining a P balance on the farm. Theoretically,this can be viewed as the minimum cost for reducing and maintaining phosphorus inthe farm’s soils at acceptable levels. This reduction in earning potential would likelybe translated into decreased agricultural use land values. Due to the high cost, theefficacy of policies and programs to control phosphorus loading on farms usingmanagement-intensive grazing must be examined relative to other dairy productionsystems. In reality, if dairy farms are constrained to achieve a P balance, farms usingmanagement-intensive grazing may be able to do so at a lower cost than other dairyproduction systems (Winsten et al., 2000b).In addition to the level of such a cost, who bears the cost, if anyone, and how such

a program is implemented are important policy questions. Estimating the value tosociety of reducing phosphorus loading is a prerequisite for informed economicdecision making regarding such policies. Regardless of where it is placed, the level ofincentive required to induce optimal producer behavior is inextricably linked to theprofitability of production. A dairy farm with lower per cow profitability will beinduced to take action by a lower incentive level, ceteris paribus, because it will costthem less to do so. The minimum incentive level required to achieve targeted soilphosphorus levels, then, would be different for each farm based on their cost struc-ture. Efficiency gains are then possible by first inducing actions from the producersfor whom it is least costly to reduce soil phosphorus levels. This point is an impor-tant motivating factor in the discussion of tradeable permits for controlling pollution.It is important for policymakers to understand the potential environmental con-

sequences of current farm income support policies. Decoupling farm support payments


from output production levels may be necessary to achieve the simultaneous objec-tives of a viable farm sector and reduced agricultural nonpoint source pollution.

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