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A pricing model for forage in British Columbia Haggard, Trenton John 1994

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A PRICING MODEL FOR FORAGE IN BRITISH COLUMBIAbyTRENTON JOHN HAGGARDB.A., The University of British Columbia, 1992A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE FACULTY OF GRADUATE STUDIES(Department of Agricultural Economics)We accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIASeptember 1994-Trenton John Haggard, 1994In presenting this thesis in partial fulfillment of therequirements for an advanced degree at the University of BritishColumbia, I agree that the Library shall make it freely availablefor reference and study. I further agree that permission forextensive copying of this thesis for scholarly purposes may begranted by the head of my department or by his or herrepresentatives. It is understood that copying or publication ofthis thesis for financial gain shall not be allowed without mywritten permission.(SignDepartment of ,ç4QjJ5 4D/ii?jThe University of British ColumbiaVancouver, CanadaDate_________iiABSTRACTThe production of forage in British Columbia plays and integral role insustaining livestock herds within the province. Forage is an important componentin the daily feed requirements of horses, sheep, and cattle. Fluctuations in theavailability of forage due to drought or bad weather conditions can imposeconsiderable costs on farmers who raise livestock. Wide—spread droughtconditions can significantly limit the availability of forage crops withincertain regions, causing prices within those regions to become inflated.Under standard insurance in British Columbia, farmers are only insuredagainst shortfalls in production; there is no compensation provided againstincreases in the price of forage. For those purchasing forage, a Wide—SpreadDrought (WSD) insurance scheme would provide insurance against the price—riskassociated with drastic weather conditions. However, since forage prices arerequired to operate such a policy and are non—observable, a mechanism is neededin order to estimate them. A regional spatial price—equilibrium model whichrelates regional prices to regional production is developed in this thesis. Themodel will eventually be used to predict prices and hence determine whether aparticular region is eligible for a payout under the WSD insurance scheme. A keyassumption behind the model is that according to the ‘Law of One Price’; pricesare perfectly arbitraged. In a competitive setting, in which agents maximizeindividual welfare, total welfare is maximized and prices between regions willnot differ by more than the transportation costs.This spatial price—equilibrium model is applied to British Columbia forageproduction. The regions incorporated in the study include the Peace River,Central Interior, Cariboo—Chilcotin, Thompson—Okanagan, and Kootenay Regions.The Lower Mainland/Fraser Valley and Vancouver Island are excluded as they do nottypically fall under the forage crop insurance plan in British Columbia.Table of Contents3.1 Methodology3.2 Demand Curve3.3 Kinked Demand Curve3.4 Spatial Aspects3.5 Storage3.6 Mathematical Model3.7 Simulation Process4. British Columbia Wide—Spread Drought Insurance SchemeForage Production : A Case Study4.0 Introduction4.1 Application4.2 Computer Algorithm5. Results6. Conclusions.7. Bibliography.8. Appendix 19. Appendix 210. Appendix 3iiipageAbstract iiTable of Contents iiiList of Figures ivList of Tables vAknowledgements vi1. Introduction 11.1 Background 11.2 Problem Statement 31.3 Study Objectives 51.4 Organization of the Study 62. Review of Literature 72.0 Summary 73. Methodology and Model 123.0 Overview1213172020212331for33333337405658606567ivList of Figurespage41.1 Wide—Spread Drought compensation payment3.1 Description of a region 133.2 Welfare maximization and the ‘Law of One Price’ 163.3 Regional demand curve for forage 183.4 Demand curve for storage 233.5 Benefit measure under upper slope of demand curve 243.6 Benefit measure for points under lower slope ofdemand curve 254.1 Regional breakdown for British Columbia 345.1 Price/quantity plot for Peace River Region(Base Case Scenario) 425.2 Price/quantity plot for Central Interior Region (Base) 435.3 Price/quantity plot for Cariboo—Chilcotin Region (Base) 435.4 Price/quantity plot for Thompson—Okanagan Region (Base) 445.5 Price/quantity plot for Kootenay Region (Base) 445.6 Predicted over real price for Peace River Region 505.7 Predicted over real price for Central Interior Region 505.8 Predicted over real price for Cariboo—Chilcotin Region 515.9 Predicted over real price for Thompson—Okanagan Region 515.10 Predicted over real price for Kootenay Region 525.11 Price/quantity plot for Peace River Region (Base Case) 55List of Tablespage1 Regional livestock numbers 612 Regional base herd feed requirement 613 Yield per acre for forage crops 624 Seeded acreage and average regional production 625 Coefficient of variation/correlation coefficient matrix forregional precipitation 636 Cholesky’s decomposed matrix for production 637 Regional transportation cost matrix 648 Regional storage costs 64vviAcknowledgementsI would like to thank the members of my committee: Dr. James Vercammen,Dr. Casey van Kooten, and Dr. W. Waters. A special thanks to Jim, my primaryadvisor, for all of his help and effort towards this thesis. Thanks also to thethe guys at the Ministry of Agriculture, Fisheries, and Foods (Crop InsuranceBranch) : Richard Scott, Lonie Stewart, Wayne Lohr, and Bob France. Without thefunding and support from the Ministry and its members, this study would not havebeen possible.Thanks to all my fellow students and members of the Department ofAgricultural Economics, for putting up with me and providing occasional insightto the thesis. A special thanks to Jodie for the use of her broad vocabulary andsuperior editing skills.Mostly, I would like to thank my family for their never—ending support andencouragement. Yeh, this thing is finally done...1Chapter 1 : Introduction1.1 BackgroundForage is an important crop in British Columbia as it contributes to theBritish Columbia livestock industry. Livestock producers use forage as one ofthe main ingredients in the composition of livestock feed. Forage in some casescomprises a significantly large portion of livestock dry matter intake (up to 70%for sheep and 100% for horses) . Percentage of forage use per dry matter intakefor cattle differs depending on type and sex of animal; however, a rough estimateof forage consumption by cattle would be approximately 30 lbs per day, 6 lbs forsheep, and 16—20 lbs for horses.’Relative to the land devoted to its production, forage is a highlysignificant crop in British Columbia.2 More than 800,000 acres was devoted toits growth in 1991, compared to slightly over 100,000 acres for each of wheat,barley, and canola. The largest allocation of land devoted to forage occurs inthe Thompson—Okanagan Region under which approximately 50 percent of the land isirrigated. Irrigated land can also be found in the Kootenay (approximately 50%of the land), Cariboo—Chilcotin (less than 50%), and Central Interior regions(under 20%), with no irrigation occurring in the Peace River Region (StatisticsCanada #95—3935, 1991)On average, the province tends to produce enough forage to meet its ownneeds. In a typical year in which average yields occur, both the Peace River andKootenay Region are relatively self—sufficient (producing enough to meet theirrequirements) . The Cariboo—Chilcotin Region produces less than it requires and,as a result, crops will flow in from the Central Interior and Thompson—Okanaganregions .1 For further information refer to Keay (1991), National Research Council (1989), AgricultureCanada (1986), and Beames et al. (1994).2 Forage refers to alfalfa and other types of hay used as a component in livestock feed.On average, these two regions produce more forage than they require.aIn British Columbia, the majority of forage crops are produced by those whoutilize it (for livestock feed) . These farmers commonly store some of theircrops for use in following seasons. However, in times of drought (periods ofunusually low levels of precipitation), this is often not enough to meet theirherd’s forage requirements. As a result, farmers will purchase forage locallyor from other regions (other parts of the province, Alberta, or Washington) ata price which is based on regional supplies and demand. Given that forage hasno close substitutes in feed use and is relatively expensive to transport betweenregions, forage demand tends to be quite inelastic. This inelastic demand,combined with highly variable yields and quality (due to variability in moistureand natural inputs in production), results in a price of forage that is rathervolatile.Since the majority of livestock producers in British Columbia are bothproducers and consumers of the crop, they are not only concerned when forageprices fall, but when they rise as well. Higher prices often relate toshortfalls in yield, and as a result, farmers must face both a reducedavailability of forage and an inflated price in making up the shortfall. A fallin the price of forage is often associated with an excess supply, and given thatfarmers typically consume the crops they produce, the situation is not as severe.Forage prices vary inversely with yield levels 6 and because wide—spread droughtscan occur as frequent as one in five years7, price—risk is an importantconsideration.is commonly transported via truck, as it is the most available and convenient methodof transport.It is not uncommon for forage prices to rise 50 percent above the average during a widespread drought.6 For example, a 20 percent decrease in yields below the average would result in a 20 percentincrease in forage prices.Obtained from yield series data from Agriculture Canada (1970-74), Tingle (1975-87),Forage Cultivar Trial Summary (1980-93).1.2 Problem StatementUnder standard crop insurance in British Columbia, farmers are onlyinsured/compensated for shortfalls in production that fall below a guaranteelevel (some percentage of average production), with the losses valued at anaverage price level. This means that compensation or indemnity payments equalthe shortfall in production below a guarantee level multiplied by the averageprice level. Farmers who must purchase forage are not well covered in ashortfall year because forage prices tend to rise in shortfall years. It hasbeen proposed by the British Columbia Ministry of Agriculture, Fisheries, andFoods that a WSD insurance scheme be designed to address this price—risk facingfarmers.Under a WSD insurance scheme, farmers would be insured against the rise inprice of forage due to wide—spread drought, with indemnity payments equalling theshortfall in production below a guarantee level multiplied by the differencebetween the market price of forage and a price trigger if the former exceeds thelatter.8 A wide—spread drought would be necessary but not sufficient to triggera payment from this scheme. This is because when adequate stocks are availablein nearby regions, stocks would flow in to alleviate the shortage and the pricein the shortfall region would not rise above the trigger.To qualify for a WSD payment, farmers would have to be eligible forstandard insurance (i.e., if their actual production falls below the guaranteelevel) and have regional forage prices exceeding a threshold price level (i.e.,exceeding some percentage of the insured value/average price) . There is,however, a problem with implementing a policy such as this one, as somemeasurement of the actual forage price is required. Since no formal marketexists for forage crops, prices are non—observable. This means that withoutsome mechanism for determining the actual price levels, the values of indemnities8 The price trigger level would be some percentage above the average price.Transactions regarding the sale of forage occur privately between farmers, and prices varydepending on factors such as quality of hay, transportation costs, and types of transactions(personal discounts between friends, bartering, etc).under this program would be unknown.The purpose of this thesis is to present a pricing model for forage. Onceconstructed, this model can then be used for the purposes of crop insurance, asa mechanism will be available for estimating forage prices. Specifically,current levels of forage supplies would be incorporated with the regressionresults from the study to generate regional forage price estimates. Given theseprice estimates and the observed production levels, the insurers can determinethe level of indemnity payments under the Wide—Spread Drought insurance scheme.Figure 1.1 represents a region for which an indemnity payment will occur. Theactual (estimated) price exceeds the price trigger level and production fallsbelow the guarantee level. The level of indemnity is shown by the shadedrectangle.PriceLevelQuantity LevelPrice exceedsEt; mate4-Price TriggerGuaranteeLevelAct&lP(a.LtShortfall inProductionFigure 1.1 Wide-Spread Drought compensation payment.51.3 Study ObjectivesThe main purpose of this study is to devise a theoretically acceptable andpotentially useful method of estimating the price of forage. Given the problemof trying to establish prices in different areas for a good that flows within andbetween these areas, incorporating spatial dimensions into the model is required.A regional rather than individual agent model is used because the WSD insurancescheme will be based on regional production and not farm—level production.Since the market for forage is assumed to be competitive and as a resultprices in the regions will not exceed the transportation costs between them, themodel utilizes the ‘Law of One Price’ assumption. The fact that farmers arerational, profit maximizers and crops are able to flow freely between regionsensures that the ‘Law of One Price’ will hold. The characteristics of eachregion will be based on representative agents within that region. Furthermore,the ability of each region to place production in storage for use in futureperiods will also be incorporated.There are two main components to the model, the first one being that givena set of observations on regional supplies of forage, it will show theequilibrium allocation for those quantities and the set of equilibrium prices.The second component allows for simulations to be run; that is, randomly drawnproduction levels can be made. The random draws explicitly account for thedifferent production variances and covariances across regions. Combining bothcomponents, production levels are drawn and equilibrium allocation levels andprices solved for. This can be done many timesin order to get a series ofequilibrium prices associated with the simulated quantities. Regression analysisis used to draw relationships between regional prices and quantities. Theseresults can then be used in an insurance scheme, where, given regional quantitiesof forage, prices can be forecasted. 1010 Although econometrics are used in this study, the model presented is a simulation modeland not an econometric one. The econometrics are done on simulated and not real data.61.4 Organization of the StudyChapter 2 provides a review of the literature and is followed in Chapter3 by the methodology used in this study. Chapter 3 continues with a descriptionof the model and the assumptions held. Chapter 4 presents an application of themodel to British Columbia forage production and includes some description of thedata used and its sources. Chapter 5 and 6 follow up with a description of theresults, conclusions, and recommendations regarding the model’s application. TheAppendix contains a description of the data used and generated, sensitivityanalysis results, and copies of the computer algorithms used.Chapter 2 : Review of Literature2.0 SummaryAssuming that the model would be of a regional, spatial allocation nature,a search of the literature was undertaken. The main focus of the search was toidentify past literature that had approached the problem of estimating forageprices (or of similar crops), and which had simulated a spatial allocation typesetting.Prior to 1984, no published studies relating to the estimation of forageprices could be found. A study by Blake and Clevenger (1984) noted the same, andfound only one unpublished study by Myer and Yanagida (1981) relating to thistopic.11 Blake and Clevenger stated that the Myer and Yanagida’s paper combinedan estimated demand function for alfalfa in 11 western states with a quarterlyARIMA model to forecast quarterly alfalfa hay prices. The Blake and Clevengerpaper, however, developed a slightly different model that forecasted monthlyalfalfa hay prices before the first harvest, for the state of New Mexico. Theyused a two step procedure that linked an annual model, forecasting the point atwhich seasonal price patterns start, to a monthly model that identified theseasonal price patterns. They incorporated the estimation of a series of monthlyautoregressive price forecasting equations, an annual alfalfa demand equation,and an annual autoregressive acreage forecasting equation. These results werethen used to predict monthly alfalfa prices for the state.In 1987, Blank and Ayer created an econometric model of the alfalfa marketfor the state of Arizona. A similar study by Konyar and Knapp (1988) providesan analysis for the aggregate California market. A later study by Konyar andKnapp (1990) incorporating much of their previous research, presents a dynamicspatial price—equilibrium model of the California alfalfa market. Their modelwas used to forecast alfalfa acreage, prices paid and received, andtransportation flows for the short and long run under base year conditions. The“This Myer and Yanagida study was later published in 1984.8base year results were then used for comparison in determining the effects ofreductions in federal water subsidies and the implementation of a cotton acreage—reduction program.There are many other studies, aside from those focusing on price estimationof agricultural crops, which have focused on spatial allocation and pricing underthe spatial allocation setting. One common assumption made in many of thesestudies is that in a competitive, spatial environment in which goods can movefreely from one agent to the next, the ‘Law of One Price’ holds. There are somestudies that may lead one to question the appropriateness of the ‘Law of OnePrice’ hypothesis, such as Ardeni (1989) which showed that some of the evidenceto support the existence of perfectly arbitraged commodity process in the longrun, is flawed due to inferior use of econometric techniques. Other studiescounter these attacks, like Baffes (1991) who states that the ‘Law of One Price’still holds and any contrary evidence relates accounting for the transactioncosts as the failure. Regardless, the ‘Law of One Price’ hypothesis will bemaintained within the current study.A competitive spatial equilibrium setting is simulated in the study by Liewand Shim (1978) . They take the theoretical problem of maximizing an arbitrarynet welfare function. It is reduced to the Dantzig—Cottle fundamental problem,which is less complicated than the simplex tableau method as additional variablesoutside of the original problem are not necessary for obtaining feasiblesolutions. They discuss the economic implications of dual, slack and surplusvectors and the welfare maximizing marginal transformation of demand and supplyamong regions. A similar concept to ‘The Law of One Price’ is assumed, in whichthe price of the kth commodity in region j should not exceed the sum of thetransportation costs required to deliver that commodity from region i to j andthe supply price of the kth commodity in region i. A numerical example of themodel is also provided.A paper by Willett (1983) incorporates a typical competitive spatial price—equilibrium model, with both a one commodity and multi—commodity settingrepresented. This is all done within a linear programming framework. The study9further examines and tests the theoretical conditions on prices and quantities,within ‘Duality Theory’, for a competitive spatial equilibrium solution to beobtained.A study by Beckmann (1985) offers an interesting look at competitivespatial pricing under two separate pricing techniques. The effects of changingtransportation costs, size of fixed costs, and consumer density on the radius ofmarkets, under both techniques is examined. Further, the effects on agents’profits and welfare were examined. Unfortunately, the majority of informationprovided within this study does not relate directly to the problem ofestablishing a competitive spatial price—equilibrium type model. Only specificeffects that changes in parameters have on the overall solutions are identified.In Takayama and Labys’ (1986) study, a general overview of analysis withina spatial allocation environment is presented. An example of a typicalinternational spatial equilibrium analysis between two countries with onecommodity is shown, followed by the general description of a typicalinterregional spatial equilibrium model. A comparison between the use of thequadratic programming method and linear complementarity programming method in astatic spatial equilibrium framework is then made. 12 Furthermore, reference ismade to some of the recent models constructed for agriculture, energy, andminerals use. These basically describe the new techniques used by some of themain agents within these sectors.Another general overview of spatial economic theory is presented in thebook by Harris and Nadji (1987). It begins with a general description of‘Spatial Theory’, then refers to spatial equilibrium models in relation to‘Location Theory’ . It explains that many of the spatial equilibrium models arepartial equilibrium special cases of the general theory. The general system asa non—equilibrium dynamic theory is described. Finally, a discussion of thetransition of the theoretical framework to an applied model is presented,including a description of the construction of and equations associated with an12 A dynamic type framework was not presented in this study.10applied location theory model.Although there have been few studies done on the pricing of forage (orsimilar products), much literature exists on the creation of competitive spatialprice—equilibrium models. The techniques used in the majority of these studiesare directly applicable to the current paper. It was established from thebeginning that a regional, competitive, spatial price—equilibrium model was tobe used, and the studies shown, provided a general basis for the modelrepresented in this study.The current study’s model typically assumes regions to be both producersand consumers of forage, and allows for forage crops to flow freely within andbetween regions depending on transport costs, availability, and regional foragerequirements. It is a result of this competitive setting that the ‘Law of OnePrice’ assumption can be made. The current model is similar to the other spatialprice—equilibrium models, especially the one used in Konyar and Knapp (1990)They assume a competitive market exists for the good and that it can flow freelybetween agents depending on supply and demand. Like the Konyar and Knapp model,the current one assumes regions to be both producers and consumers of the crop.There are, however, a few notable differences that will be outlined.A dynamic model is used in the Konyar and Knapp study, in which there isa direct link between individual periods. The Konyar and Knapp model containsan acreage response characteristic, where major producing regions havefluctuating acreage depending on acreage from previous periods, expected pricesreceived, and yields. The current study does not make any reference to acreageresponse, as individual agents are assumed to be price takers and produce forage,independent of expected prices. Making no direct link between periods and havingno acreage response characteristic, the current model is not truly dynamic.Carry—over stocks are included in the study but each period is treated asindependent of the others.The Konyar and Knapp study, like the majority of spatial price—equilibriumstudies assumes linear inverse demand curves for the good in question, forsimplicity. This may be a model mispecification when modelling forage11production. Since farmers have specific base feed requirements to meet, andthere are costs to adjusting herd sizes, the individual farmers’ demand forforage will not fluctuate given small price variability around an average pricelevel. Therefore, an inelastic portion to the demand curve is needed around theaverage price level when modelling individual farmers. A regional demand curve,however, will not necessarily have the immediate upper and lower kink in thedemand and may be slightly smoothed. 13 Nonetheless, kinked regional demandcurves were incorporated into the current study, capturing the reluctance offarmers to alter their base herd size.The current study, unlike the others, uses a simulation and not aneconometric model. Production data is randomly generated, with the optimalspatial allocation of supplies and their associated prices calculated.Econometrics is then used on these results in order to formulate a pricingmechanism. In the Konyar and Knapp study, actual production data are used. Theyalso use econometrics in creating the pricing model, however, their results arenot simulation—generated as in the current study. Further, the Konyar and Knappmodel does not exploit the covariability between regional production (due tocommon weather patterns)The aforementioned characteristics of the current study provide somepotentially useful techniques which can be added to the past body of researchdevoted to pricing forage. Studies relating to this topic are few andinformation regarding the prices of forage can benefit those involved in theproduction of it and those involved in providing crop insurance and other typesof government assistance.13 When the representation of forage production is more diverse within a region, the regionaldemand curve may contain a smoother upper and lower kink.12Chapter 3 : Methodology and Model3.0 OverviewThis chapter introduces a pricing model for forage. The model can bebroken down into two main parts: one part generates random regional productionand finds the optimal allocation of that production, and the second part performsthis over numerous simulations so that a series of quantities and theirassociated prices can be created. Regression analysis is then used to drawrelationships between the quantities and the prices. A more detailed descriptionof these parts is described below.The first portion of the model allows for regional production levels to besimulated. This is done by randomly drawing regional production levels fromnormal distributions around their means. 14 Carry—over stocks are added to therandomly drawn production levels to create regional supplies. Given these supplylevels, the model optimally allocates quantities for an equilibrium solution.This solution is reached by assuming that each region maximizes its welfare givenits own demand and supply for forage, prices in other regions, and transportationcosts. As a result, crops will flow within and between regions in order thattotal welfare be maximized. 15The second part of the model randomly draws production levels and solvesfor equilibrium solutions over a number of simulations. The random quantitylevels and their associated prices (as determined by points on the regionaldemand curves) are collected from each simulation such that a series ofquantities and prices are generated.16 Econometrics is then used toparameterize the relationship between regional quantities of forage and the14 Normal distributions are used in the random draws, since the associated data requirementsare small and multivariate random normal draws are more easily obtained than those with otherdistributions.15 Total welfare equals the sum of each region’s welfare.16 Prices are also considered to be normally distributed.13associated regional prices3.1 MethodologyThe model analyzes transportation flows and price fluctuations at aregional level. Each region is described as being both a producer and consumerof forage, and is characterized by a representative agent in that region. It isassumed that the representative farmer produces forage in order to feed his/herown base livestock herd. The regional livestock feed requirements depend on thenumber of livestock present and the feed requirements per animal.From figure 3.1 shown below, a typical region has both a production andconsumption sector and, depending on current supplies and demand has a number ofoptions in order to meet feed requirements and maximize welfare. If surpluscrops are present, the region can allocate stocks to storage for future use orship to other regions. In times of excess demand, stocks can be drawn fromstorage (if they are present) or shipped in from other regions. The arrows showthe direction in which forage crops will flow.[ Typic& Region 1nuirather Region Storage RegiFigure 3.1 Description of region14Farmers are assumed to be rational profit maximizers. They have a specificbase herd size, and will produce forage in order to meet the feed requirements.When a shortfall in production occurs such that the base requirement cannot bemet, the farmer will consider purchasing forage to make up the shortfall. Thereverse happens when there are surplus stocks. If the base requirement is met,the excess stocks will either be placed into storage for future use or be sold.Only if the price rises to a sufficiently high level or falls to a sufficientlylow level will the farmer move away from his/her base herd requirements.Farm level behaviour must be assumed, since this is a regional model anddoes not explicitly observe farm level actions. Farmers are considered to beprofit maximizers, facing parametric prices. They buy and sell forage in orderto maximize their individual profits and as a result, a competitive market isestablished where total welfare is maximized. Since all farmers are maximizingindividual profits (welfare), and total welfare is defined as the sum of allindividual welfares, total welfare is maximized. This assumption of welfaremaximization allows for the creation of an equilibrium setting in which themodel’s key assumption, the ‘Law of One Price’, may hold.The ‘Law of One Price’ states that the price in any one region will neverexceed the price in another region by more than the transportation costs. It isthe assumption of a competitive market that validates the ‘Law of One Price’ inthis model. In a competitive world, individual agents seek to maximize their ownwelfare, and the collective action of all agents can and will affect prices.Given an arbitrarily high price in one region (i.e., price exceeding that ofanother region by more than the costs of transportation), individual agents (andas a result the collective of agents) will arbitrage on this high price. As aresult, the price will fall until arbitrage is no longer feasible and the ‘Lawof One Price’ holds. Therefore, it can be stated that when individuals maximizewelfare the ‘Law of One Price’ holds, and when the ‘Law of One Price’ holds,total welfare is being maximized. If the ‘Law of One Price’ is not holding, thenindividuals are not profit maximizing and total welfare is not being maximized.The following diagram, Figure 3.2, presents examples which help to validate15that the ‘Law of One Price’ will hold when welfare is maximized. The firstexample occurs at point A for both of the regions. In this case, no trade takesplace, with Region 1 producing and consuming at the point CAl and Region 2producing and consuming at point CA2. Assume that the ‘Law of One Price’ isviolated, where the price in Region 1, PAl, exceeds the price in Region 2, PA2,by more than the costs of transportation. Since an autarky example is beingrepresented, total welfare can be determined by strictly looking at consumptionlevels under the value of marginal product curves (demand curves) . Region 1consumes at CAl, therefore its welfare can be measured by the area under itscurve, areas 1, 2, and 3. Region 2 consumes at CA2, therefore its welfare isshown by areas 7, 8, 9, 10, and 11. The total welfare is measured as the sum ofthese two welfares.When the regions are able to trade, crops will flow from Region 2 to Region1, since the agents can be made better off by this. Trade will take place untilthe point at which the ‘Law of One Price’ is no longer violated (the point wherethe price in region 1, PB1, exactly exceeds the price in 2, PB2, by thetransportation costs) . It is at this point that trade can no longer make bothregions better off, since the price for which the crop is sold is equivalent toits marginal value in consumption. ‘ It is at this point that total welfare ismaximized, since any other allocation of crops other than this equilibriumallocation will cause total welfare to decrease.The total welfare associated with the second case has to include both thevalue from sale and from consumption, since trade has occurred. Region 1purchases forage from Region 2 and increases its consumption to the point CB1.The cost of that purchase is equal to the price paid, PB1, multiplied by thequantity difference between CAl and CB1. This is a cost shown by area 5.However, Region 1 now benefits from areas 4 and 5, and therefore, gains area 4.The sale of forage from Region 2 to Region 1 means that 2 now consumes at thelower level of CB2, and therefore, loses the consumption welfare shown by areas17 Note that when crops flow from Region 2 to 1, the price in 1 will fall (as supply increases)and the price in 2 will rise (as supply decreases).1610 and 11. However, the sale of forage benefits Region 2 by the value of PB2(the price sold at) multiplied by the difference between CA2 and CB2. As aresult, Region 2 gains a welfare amount equal to the shaded area above the demandcurve. For a linear curve this welfare amount is equivalent to area 10.When the two regions are able to trade, the allocation of crops between theregions will be such that total welfare will be maximized, and the ‘Law of OnePrice’ will hold. In this example, both regions are made better of by tradingto the point that prices mc longer differ by the costs of transportation.Compared with the first example, Region 1 shows an increase in welfare equivalentto area 4, and Region 2 gains an amount equivalent to area 10.Region 2Figure 3.2 Welfare maximization and the Law of One Price’.The previous example relates directly to the situation in which a droughtin one region creates a shortage of crop and causes the price in that region toRegion 1PricePAlPB1DemandCurvePriceP82PA2DemandCurveOA1 CB1 Consumption CB2 c12 Consumption17increase. Due to arbitrage from welfare maximizing agents, surplus regions willship crops into the drought region and reduce the level by which that region’sprice will increase. Since crops are flowing out of the surplus regions into thedrought region, the surplus regions’ prices will rise and the drought region’sprice will fall. As explained earlier, flows of crops will occur to point atwhich prices in all regions will not differ by more than the transportationcosts.3.2 Demand CurveEach region has its own demand curve for forage which represents the valueplaced on forage in that region given regional prices. This value is representedby the areas under the curves (the cumulative value of each unit of forage) . Forgiven quantities of forage, regional prices can be established by the respectivepoints on the curve.The demand curve is a value of marginal product curve, and points on thiscurve refer to points where the price of the marginal unit of forage equals itsvalue of marginal product in production of livestock. Therefore, when agentsmaximize welfare, they will purchase forage up until the point where the priceof the last unit of forage equals the value of its marginal product. If agentsdo not purchase forage up until this point, welfare will be measured at somepoint below the demand curve, where the price of a marginal unit is lower thanits value of marginal product. In this case the agents would not be maximizingtheir own welfare. If all agents maximize welfare then total welfare ismaximized and regional prices can be determined by the respective points on theregional demand curves.The shapes of regular demand curves are typically assumed to be continuousand downward sloping. However, when modelling forage this is not an appropriateshape of curve to use. Since forage crops have a lack of close substitutes, andfarmers do not readily alter base herd sizes given small fluctuations in forageprices, the shape of demand curve is not necessarily continuous and downwardsloping.18Referring to figure 3.3, the shape of a regional demand curve will beperfectly inelastic (vertical) for a given range of reasonable forage prices.Since farmers have specific stock requirements which they must meet, and thereare costs to adjusting herd size (i.e., actual physical costs of buying andselling cattle and the uncertainty associated with it), the demand for foragewill not fluctuate given low variability in prices around an average price level.The height of this vertical portion of the demand curve will depend on thedistribution and scale of the different farms in the region. If the majority offarmers are operating at the margin, then the vertical portion may kink soonerat the top. These types of farmers will be more responsive to increases in hayprices since their scale of operation is not as profitable (flexible) at themargin as other farms. If the scale of farms in the region is widelydistributed, then the upper kink may become smooth as the farms at the marginrespond to small price fluctuations and the other farms gradually respond tolarger price fluctuations.Price ofForageUpper KinkPointInelastic Portionof CurveLower KinkBase Herd Feed Forage ConsumptionRequirementFigure 3.3 Regional Demand Curve for Forage19The regional quantity demanded for forage will likely respond if forageprices become excessively high. Under a wide—spread drought scenario, with belowaverage precipitation across a large area causing decreased forage yields, theremay not be adequate stocks in nearby regions to meet the excess demand for foragein that region. As a result, forage prices may become so high that farmers beginto decrease their consumption of forage. This would occur when it is no longerfeasible for farmers to maintain and feed their present base herd size given thehigher forage prices. The result is a decrease in the demand for forage asfarmers seek alternatives (i.e., reduce their base herd size; use more of othergrains like barley, if possible; sell off feeder calves earlier than expected;and feed less hay per animal) . If the drought is more severe and prices rise toa higher level, the demand for forage will be even less and the demand curve willcontinue to slope backward to the left.The demand curve may be kinked at the bottom of the vertical portion sinceexceedingly low forage prices may entice farmers to increase their base herd sizeand forage usage. Given farmers present base herd size and feed requirement (asrepresented by the vertical portion of the demand curve) when forage prices falllow enough (eg., favourable weather patterns across large areas causing surplusforage yields and large drops in forage prices), farmers may choose to invest inlivestock. Given the drop in prices, farmers can meet the feed requirements ofa base herd larger than the present herd. Although this bottom kink can bedebated, it is the upper kink and slope that are of interest for the model (theprice responses to drought situations on the upper portion of the demand curve)A graphical description of a regional demand curve would assume a linearcurve intercepting the vertical axis and sloping downward to the right. At thebase herd feed requirement level (as described on the horizontal axis), thisdemand curve becomes downwardly vertical. This represents to the inelasticportion. At the bottom of the inelastic segment, the demand curve slopes downtoward the right again.203.3 Kinked Demand CurveWhen using a continuous downward sloping demand curve, it is not necessaryto know the section of curve being utilized because one function represents allpoints on the curve. However, non—continuous (kinked) demand curves require thatthe section of curve be identified. This is due to a separate linear functionused for both the upper and lower slopes of the curve. If optimization occursto the left of the inelastic portion, then the upper part of the curve is used,and if it occurs to the right of the inelastic portion, then the lower part ofthe curve is used. The upper portion of the regional demand curves containseparate parameters than the lower curves. The upper portion is linear anddownward sloping. In mathematical form it can be described as a verticalintercept (au, the subscript ‘u’ referring to the upper curve), plus a negativeslope value () multiplied by the level of consumption (C, represented on thehorizontal axis) . The lower portion of the curve is described as the verticalintercept (c’, the subscript ‘1’ referring to the lower curve), minus a slope(1) multiplied by the level of consumption (C) . Note, that the method used inidentifying which section of the curve is being referred to is later describedin a mathematical description of the model.3.4 Spatial AspectsAs discussed previously, the model allows regions to allocate foragesupplies within and between themselves, and to and from storage, in order tomaximize welfare. When dealing with continuous downward sloping demand curvesin this type of spatial setting, the results will appear to be uniform(price/quantity relationships are based on continuous curves) . However, whenincorporating non—continuous (kinked) demand curves like those in this study, theresults will differ (price/quantity relationships are based on non—continuouscurves)When regions optimize welfare under a kinked demand curve, the resultscontain jumps when moving from one section to the other in the demand curve. Thevertical intercept (and possibly the slope) are different when referring to the21two separate portions of the curve to the left and to the right of the inelasticsection. This inconsistency in the demand curve will create different types ofsolutions in which the ‘Law of One Price’ holds. In the optimization process,regions will not have the gradually decreasing marginal benefit as consumptionincreases. Marginal benefit does gradually decrease to a point, as consumptionincreases; however, an increase in consumption beyond the base herd feedrequirement results in a discontinuous fall in the marginal benefit (moving fromthe upper slope to the lower one)Regional prices may also exhibit some non—uniform results. When regionalconsumption levels are to the left of the inelastic portion, pricing is based onthe upper slope. And, when consumption is to the right of the inelastic portion,prices are based on the lower slope. But, when regional consumption exactlyequals the regional base herd feed requirement, pricing cannot be based on theregion’s demand curve, as the actual price level is somewhere within theinelastic portion. It is known from the ‘Law of One Price’ that when regionstrade, flows will occur to the point that regional prices will no longer differby transportation costs. Therefore, the region whose consumption equals its baseherd feed requirement can calculate its price from another trading partner’sprice, plus or minus transportation costs depending on the direction of flow.If the region is importing forage, then its price is equal to the other region’sprice plus transportation costs, and when exporting forage, its price equals theother’s price minus transportation costs.3.5 StorageSome dynamic characteristics are incorporated into the model. Thesecharacteristics show that farmers at most will look one period ahead into thefuture, by allocating stocks to storage for use in that period. Given farmers’profit maximizing behaviour, they will arbitrage on forage prices through theiruse of storage. Assuming that farmers are rational, they will determine whetheror not to store production for use in the following period given current forageprices, costs of storing across one period, and the expectation that next period22will witness an average price given average production.Not only are farmers able to send forage stocks into storage, but they alsoreceive stocks from storage. Therefore, when farmers determine how to optimallyallocate current production, they include the amount of stocks they are receivingfrom storage during that period. The model forms no continuous link across timebetween the amount of forage shipped to storage in one period and the amount ofstocks in storage added to production in the following period, since each periodis treated as an independent simulation. Given the model has no formaldependency across time, it cannot be considered a truly dynamic model, however,the aforementioned aspects relating to storage do provide some mildly dynamiccharacteristics to the model.A steady—state characteristic is necessary when including storage in thismodel. On the average, the carry—over stocks included in a current period’ssupply must equal the stocks allocated for use in the following period. Withoutthis characteristic, the results will be biased as current supplies will notequal their true average values. Steady—state for storage is incorporated intothe model by running the set of simulations a number of times, until on averagestocks in equal stocks out.’9Figure 3.4, presents a graphical representation of the demand curve forstorage. The expected price next period is represented on the vertical axis withstocks allocated to storage for use next period on the horizontal axis. Thecurve intercepts the vertical axis and is linearly downward sloping. Aninterpretation of this curve would be that if no stocks are stored for the periodfollowing, an average price can be expected for next period (referring to thepoint touching the vertical axis) . As more and more stocks are allocated for thefollowing period, the expected price next period will fall further and furtherbelow the average price. This relates to the curve’s negative slope.18 Note that current forage supplies include carry-over stocks from the previous period.19 Carry-over stocks are not included in the first set of runs since they must be generated apriori. After the first set of runs are completed, stocks in are included in current supplies forallocation purposes.23Price infotlowingperiod3.6 Mathematical ModelThe model in this study determines the optimal inter/intra—regionalallocations of forage crops (for the purposes of consumption) in order tomaximize total welfare across regions. This allocation process is based onregional forage supplies and demand and on the assumption that an objectivefunction is being maximized subject to various constraints. 20 This objectivefunction represents total welfare across all regions, where total benefits (i.e.,the cumulative area under all regional demand curves) are subtracted from totalcosts (i.e., the sum of all transportation costs associated with crop flows)The following section presents a description of the model, including theobjective function and constraints.One of the main purposes of the model is to determine optimal allocations20 parameters for the regional demand curves are fixed throughout the optimization, whileregional production for each scenario is randomly drawn ex ante.Next period’sprice driven tozeroFigure 3.4 Demand Cutve for StorageQuantity stored for use nextperiod24of crops within and across regions. This is done by maximizing an objectivefunction (representing total welfare) subject to various necessary constraints.The first portion of the objective function (representing total benefits)includes mathematical formulas describing the regional demand curves. Asdescribed in the previous section, in order to incorporate the regional demandcurves into the optimization process, it is necessary to identify to whichportions of the demand curves are being referred. Unfortunately, this is not asimple task, as will be shown.When referring to a point left of the inelastic slope, the area under thecurve (welfare benefit) is measured up to the actual level of consumption (C),as represented on the horizontal axis. This area can be calculated in twoportions: the triangular area above the actual price (a,_r3*C) and the squarearea below that price. For a graphical description, see Figure 3.5 below.ConsumptionPricexu- f3 *U UActualPriceSlope = f3ActualConsumptionBase Herd FeedRequirementCFigure 3.5 Benefit measure under upper slope of demand curve25(3.7.1)The following formula represents that area (benefit) under the curve:Q5* [ (a_(a_*C) ) ] *c+ (a_*C) *C.This is equal to one—half the difference between the intercept price and actualprice, multiplied by the consumption level, plus the actual price multiplied bythe consumption level.When referring to a point right of the inelastic slope, the area (benefit)is also measured up to the actual level of consumption. It includes all of thearea to the left of the inelastic portion plus the area between the actualconsumption and base herd feed requirement level. See Figure 3.6 for a graphicalrepresentation.PriceCLcx1- 13 *CBa1--cActualPrice13’Base Herd FeedRequirementActualCB C ConsumptionFigure 3.6 Benefit measure for points under lower slope of demand curve26The area left of the inelastic portion is represented by the previousformula, with the actual consumption level (C) replaced by the base herd feedrequirement level (CB) . The area under the lower slope of the curve, between theinelastic portion and the actual consumption level, is represented by theformula:(3.7.2) Q5*[(1i*CB)_(_13i*C)]*(C_CB)+ a1*C)*(C B)This is equal to one—half the difference between the lower kink price and theactual price, multiplied by the level at which consumption exceeds the baserequirement, plus the actual price multiplied by that quantity difference.Therefore, the total area left of the inelastic portion plus the area includedto the right is represented by the formula:(3.7.3) *CBQ5**CB2+(CCB)*[aj_1(C+CB)/2)JGiven the demand curve has two distinct sections ((i.e., the portion leftand the portion right of the inelastic slope), it is necessary to identify inwhich section actual consumption falls when welfare is being optimized. Usingthe two previously described formulas for measuring benefit under the upper andthe lower slopes of the curve, identifier parameters can be used for identifyingwhere actual consumption falls. Attached to the previously defined formulas inthe objective function are two identifier parameters ‘TA’ and ‘TB’ . ‘IA’ ismultiplied by the formula relating to area measurement for points left of theinelastic section, and ‘TB’ is multiplied by the formula relating to areameasurement for points to the right of the inelastic section.The two identifier parameters are unique in that they only represent avalue of one or zero, with neither having the same value. In other words, if‘TA’ equals one, then ‘TB’ equals zero, and only the formula relating to aconsumption level less than the base herd feed requirement level is representedin the objective function. If ‘TB’ equals one, then ‘TA’ equals zero, and only27the formula for a point to the right is represented. These identifiers aresuccessfully able to identify whether consumption falls below or above the baseherd level by a series of constraints. Included in these constraints are thatthe identifiers must be non—negative, not exceed a value of one, and sum to one.Two further constraints are that ‘IA’ must be less than or equal to zero and ‘IB’must be greater than or equal to zero when consumption exceeds the base level.The reverse is true for consumption falling below the base level.These two formulas (measuring area under the demand curves) and theassociated identifier parameters ‘IA’ and ‘IB’ are included under the benefitsportion of the objective function and are summed over all regions. Thisrepresents the total benefits across all regions. The following formularepresents this portion of the objective function, in summation notation:(3.7.4)(C+CB) /2))].Note, that the subscript ‘u’ refers to the demand curve left of the inelasticportion, ‘1’ refers to the curve right of that portion, and ‘n’ refers to theregion.The second part of the objective function relates to the costs incurredthrough the allocation (transportation) of forage crops, within and betweenregions. These costs are equal to the per unit transportation costs amongregions (represented as a matrix) multiplied by the quantity levels transported(also in matrix notation) . When these two matrices are multiplied, theyrepresent the transportation costs matrix shown in the objective function, wherethe costs are represented by all of the trace elements.The transportation costs matrix is calculated by multiplying the squarematrix of per unit transportation costs (A) by the transpose of a square matrixof transportation quantity levels (T’) . The per unit transportation cost matrixis represented by each region for both the rows and the columns. This allows forall possible transportation combinations to occur, inciuding allocations within28regions. The transpose of the quantity transported matrix is used since itscombination with the per unit cost matrix yields a transportation cost matrix inwhich the elements correctly match up. In other words, the correct per unitcosts are attached to the their respective quantity levels, thereby allowing forthe costs to be represented as the trace elements in the transportation costsmatrix. In matrix notation, this transportation costs matrix can be representedby the following formula:(3.7.5) trace (A*T’),where both the per unit cost matrix and quantity level matrix are of dimensions‘n’ by ‘n’.Given the previously shown equations, the complete form of the objectivefunction can be stated in the following formula:(3.7. 6) W = Er[IA* (*C_O . 5* *C2)+IB* (a *CBO 5* *CB2+(C—CB) *in (C+CBr) /2) ) I — trace (A*T’This function represents total welfare across all regions, where the parameterW refers to welfare. It is separated into the sum of all benefits minus the sumof all costs. In the optimization process, this function (total welfare) ismaximized subject to a number of constraints, which will be described below.There are ten constraints in the optimization process. These include aconstraint that all of the choice parameters (regional consumption,transportation levels, and the two identifier parameters) be non—negative. Thisis represented as follows:(1) C, T, IAN, IBwhere consumption (C), transportation levels (T), and the identifier parametersare included for all regions.29The second and third constraints are necessary for the allocation process.These include the constraint that a region cannot consume more forage than itpossesses and gets shipped in from elsewhere. The following represents thisconstraint:(2) C, T1+T2. . .+T,where the first subscript refers to the destination region for the crop beingtransported and the second refers to the source of that crop.The third constraint relates to regions not being able to allocate moreforage than they possess (including crops shipped in from elsewhere) . This isshown by following formula:(3) QP T1+T2..where QP refers to a specific regions supply level (production and carry—overstocks)The fourth and fifth constraints relate to shipments to and from storage.These are shown by the formulas:(4) 0 and(5) T, = 0.The former states that only non—negative quantities can be allocated to storage,while the latter states that quantities cannot be allocated from storage. Thisallocation process is included separately in the program, as describedpreviously.The remaining constraints allow for the identifier parameters to function.As explained earlier, two of the constraints ensure that the identifier ‘IA’takes on a value of one or zero when consumption falls below the base level, and‘IB’ takes on a value of one or zero when consumption exceeds the base level.30These constraints are represented as following:(6) IA* (C—CB) 0, and(7) IB*(C_CB) 0.Another constraint ensures that the identifiers sum to one. This is shown by:(8) IA+IB = 1.The last two constraints force the identifiers to assume values not greater thanone. They are represented by:(9) IA, 1 and(10) IB 1.For further reference, a summary list of the parameters used in the programis shown below:W = total welfare= vertical intercept for upper slope of demand curve= vertical intercept for lower slope of demand curve13 slope of upper portion of demand curve13 = slope of lower portion of demand curveCB = regional base herd feed requirement (horizontal value of inelasticportion of demand curve)C = actual regional consumption of forageQP = regional supply of forage (includes production and carry—overstocks)A = vector of transportation costs within and between regionsT = transportation quantity levels of forageIA = identifier parameter for a point to the left of the inelastic slope(assumes a value of 1 if upper slope of demand curve is used and 0if not)31lB = identifier parameter for a point to the right of the inelastic slope(assumes a value of 1 if lower slope of demand curve is used and 0if not)3.7 Simulation PzocessAs described previously, the model used in this study is a simulation modeland not an econometric one. It randomly generates data by which the modelperforms its numerous simulations. A description of this simulation procedurewill be presented below.The model begins by generating random correlated production levels for theregions. To get these values, the model generates production levels for theregions, assuming no correlation between regional production. To do this, avector of average production levels of all the regions is added to a vector ofindependent, multivariate normal draws (with means of zero and variances of 1)This allows random independent quantities to be generated, with the mean valuesfor each region’s production taken into account.The correlations between the random production draws is accounted for bymultiplying this generated vector of regional production levels by a Choleskydecomposed matrix.21 The Cholesky decomposed matrix is a nonsingular triangularmatrix that has the property that when multiplied by a vector of independentrandom normal draws will create a vector of correlated random draws. TheCholesky matrix is created by decomposing the variance/covariance matrix ofregional production. In simplicity, the variance/ covariance matrix is createdfrom a data series of annual regional production levels. From this matrix, amatrix of characteristic vectors and diagnol matrix of characteristic roots canbe found. The two later matrices are multiplied, and together, have identicalproperties to the nonsingular triangular matrix described above.Given these random draws for regional production, the model finds theoptimal allocations for forage. This is done by maximizing the previous21 See Judge (1988),p.494-6.32objective function subject to the constraints. Note that the ‘Law of One Price’is assumed throughout this optimization procedure, since welfare maximizationensures that this law holds. The determined regional consumption levels are thenrelated to the regional demand curves to obtain the regional prices.This procedure is performed over numerous simulations in order to obtaina series of regional prices and their associated quantities. Regression analysis(OLS) is then used to draw relationships between the prices and quantities.These results can further be used for the purposes of estimating regional pricesgiven actual quantities.This pricing model has application for use in a Wide—Spread Droughtinsurance scheme for British Columbia forage producers. As noted in theintroduction, British Columbia forage producers face price risk associated withwide—spread drought. It is for this reason that a Wide—Spread Drought insurancescheme has been proposed, and subsequently a pricing mechanism needed. Thefollowing chapter will present an application of this model to the pricing offorage in British Columbia for the purposes of a Wide—Spread Drought insurancescheme.33Chapter 4 Model specification for British Columbia4.0 IntroductionThe following chapter presents an application of the pricing model toBritish Columbia forage. Included are a breakdown of the regions, a descriptionof the data used, and an explanation of the computer algorithms to perform thesimulations. The following two chapters present some of the results obtained andthe conclusions regarding the model’s application.4.1 ApplicationThe province of British Columbia was broken down into five separateregions: Peace River Region; Central Interior Region; Cariboo—Chilcotin Region;Thompson—Okanagan Region; and Kootenay Region, with each region considered botha producer and consumer of forage. This regional breakdown was determined byBritish Columbia Ministry of Agriculture, Fisheries and Foods (the study’sprimary funding agents) and based on the regional boundaries defined inStatistics Canada #95—393D (1991) . Figure 4.1 shows a rough approximation of theboundaries for British Columbia.Piovirice of BnLish Columbia34Figure 4.1 Regional breakdown for British ColumbiaThe Peace River Region includes the Peace River Regional District. TheCentral Interior Region includes both the Bulkley—Nechako Regional District andFraser—Fort George Regional District. The Cariboo—Chilcotin Region includes theCariboo Regional District. The Thompson—Okanagan Region includes the Squamish—Lillooet Regional District, Thompson—Nicola Regional District, Okanagan—Similkameen Regional District, Central and North Okanagan Regional District,Colurnbia—Shuswap Regional District, and Kootenay Boundary (Subd. B) . TheKootenay Region includes the Central and East Kootenay Regional District, andKootenay Boundary (Subd. A) 2222 The model did not include any regions outside of British Columbia. Some forage istransported between Alberta and the Peace River and Kootenay Region; however, that link wasexcluded since British Columbia is a net exporter of forage (droughts in Alberta not having asmuch impact on forage prices in B.C. than if B.C. was a net importer) and the inclusion ofAlberta would considerably complicate the problem (more regions to include, data to collect, andvariables for which to solve).35Although the model in this study does generate its own data for simulationpurposes, there are some actual data requirements that must be met in order forthe model to become operational. These requirements refer to data on demand forforage, production of forage, transportation costs within and between regions,and the costs of storage. The data and its sources are listed below.Information relating to demand for forage was needed for the study. It wasassumed that regional consumption (demand) of forage is determined by theregional base herd feed requirements. These feed requirements are based in turnon both the regional livestock numbers and feed requirements per animal (forwhich data were collected)The livestock numbers included the provinces main forage consuming animals:cattle, horses, and sheep (lambs) . The regional numbers are found in Table 1 ofthe Appendix and were obtained from Statistics Canada #95—393D (1991) . Feedrequirements for each category of animal are found in Table 2 and were obtainedfrom: Keay (1991); Agriculture Canada (1986); Ross (1989); and National ResearchCouncil (1989) . This table also includes the regional base herd feed requirementvalues used in the study.Further information on demand for forage was needed in order toparameterize the regional demand curves. Average regional long—term priceestimates for forage were obtained from various forage experts around theprovince, see Aumack et al. (1994) . These same individuals provided insight intothe responses of average farmers to fluctuations in forage prices. It was fromthese rough estimates and information that the model’s parameters werecalibrated. 23Production data was also needed. This related to information describingaverage regional production and the variance/covariance of production. Averageregional production was calculated from data characterising regional yields per23 No series data on forage consumption was available to obtain graphical estimates of thedemand curves.36acre and seeded acreage. The yield data is shown in Table 3 and was obtainedfrom: Statistics Canada #22—201 Annual Statistics and Grain Trade of Canada(1990), Statistics Canada #22—002 (1991), British Columbia Ministry ofAgriculture, Fisheries and Foods (1994). Seeded acreage data is found in Table4 and was acquired from Statistics Canada #95—393D (1991) . Also, included is theaverage regional production values used in the study.There was no series data available on regional yields, and therefore nodirect means for calculating production variability. As a result, some othermeans was needed for calculating the variance/covariance in regional yields. Anassumption that the variance/covariance in regional precipitation affectedregional yields was made. This assumption was relatively realistic sincefluctuations in precipitation are the main impetus behind variability in yields.Therefore, given series of monthly regional precipitation levels, coefficientsof variation and correlation coefficients for regional precipitation werecalculated. Assuming a relationship of 0.3 between the variation inprecipitation and variation in yield24, a variance/covariance matrix wasestablished for the regions. The variance/covariance matrix was then transformedvia the Cholesky matrix decomposition method into a nonsingular triangular matrixp, which was then used for randomly drawing regional production. 25 Thecoefficient of variation/ correlation coefficient matrix for the rainfall datais found in Table 5 and the triangular decomposed matrix is in Table 6.Transportation costs were also a necessary data requirement. A matrix oftransportation costs per ton of forage within and between regions are found inTable 7. These figures were calculated using transport cost quotes from varioustrucking companies throughout British Columbia, see McConughy at al. (1994) . Theaverage distances between regional centres is found in Ministry of Tourism(1987) . The regional centres were chosen by the Crop Insurance Branch at BritishNo studies or data were available to make this exact relationship. The value of 0.3 wasused since it appeared reasonable in comparison to other values, and yielded acceptable results.The Cholesky matrix decomposition method is found in Judge (1988) and White (1993).37Columbia Ministry of Agriculture, Fisheries and Foods, and represent the mostintense areas in each region for production and consumption of forage. Theregional centres are as follows: Fort St. John (Peace River Region); Vanderhoof(Central Interior Region); Williams Lake (Cariboo—Chilcotin Region); Kamloops(Thompson—Okanagan Region); and Cranbrook (Kootenay Region)Storage and carry—over stocks are also included in the model, therefore,information on actual physical costs associated with storage are a necessary datarequirement. Storage costs relate to the value of stored forage lost due tospoilage. The costs included in the study are shown in Table 8 and represent apercentage of the stored forage that is lost from spoilage, multiplied by thevalue of that forage. Information on the costs of storage were obtained fromBritish Columbia Ministry (1994) and Soder (1976)4.2 Computer AlgorithmIn this section, the computer algorithms in the study are explained. Themain portion of the computer calculations were made using the GAIVIS computerpackage, see Appendix 3. The program starts by defining three separate files inwhich the output is sent. Three sets are then defined, labelling the fiveregions, the number of iterations or simulations to complete, and a set used ina loop to solve for price. The parameters of the model describing the regionaldemand curves, base herd feed requirement, and mean regional production areentered using a series of ‘parameter’ commands.The parameter ‘IND’ and the Cholesky decomposed matrix2 used in randomlydrawing production levels, are entered. Two separate set of parameterdefinitions help to generate the random production levels. The first definitionassumes that the random production levels for the regions are equal to theCholesky matrix multiplied by a vector of non—correlated normal random draws(with mean 0 and variance 1), added to the mean production levels. The secondassumes that positive production levels are represented.26 Calculated previously using the SHAZAM computer package, see White (19(1993). Note:this algorithm is also found in Appendix 3.38The parameters Si through S5 are used in storing carry—over stocks to beincluded in the next 150 simulations.27 These simulations are repeated sixtimes in order for current supply levels to converge to their true values(steady—state to be imposed on carry—over stocks) . A matrix of thetransportation costs within and between regions is then entered. The objectivefunction and constraints described previously are entered, followed by adefinition of the model ‘Versioni’ and the solve command. The allocations tostorage (for use in the following period) are saved in the parameter Si and usedin the second set of 150 simulations. This is repeated a number of times for theconvergence described previously.The end of the program contains a set of ‘if—then’ statements to solve forprices. If the optimal consumption level for a region does not equal the baseherd requirement, then price is based on the demand curve. If consumption doesequal the base requirement (i.e., it is on the inelastic portion of the demandcurve), then price is based on another trading region’s price plus or minustransportation costs, depending on the direction of flow. The final set of ‘put’statements send the regional supply levels and their associated prices to theoutput files.Given the generated price and quantity series, regression analysis is usedto define the parameters for estimating forage prices from regional supplies offorage. Prices are estimated for each region under each of the 150 simulationsby substituting the regional quantities back into the regression equations.These estimated prices can then be used in determining levels of payments underthe proposed Wide—Spread Drought insurance scheme.Since the slopes of the upper portion of the demand curves are not knownwith certainty and play an integral role in the cost of the program 28, three27 Carry-over stocks are not included in the first set of 150 simulations, but are includedthereafter.The steepness of the slope of the upper portion relates changes in consumption to changesin price and thus affects the estimated price levels and the cost of the program (i.e., an increase39different slopes are used to see the effect on prices. It is assumed that witha steeper upper slope, the level of estimated prices will increase, thusincreasing the frequency and level of payments.in the level of estimated prices will increase the frequency and level of payments under theinsurance scheme).40Chapter 5 ResultsThe first set of results were obtained with all of the upper slopes of thedemand curves set to the same level. This was called the ‘Base Case Scenario’Since a means of determining the actual slopes of the regional demand curves wasunavailable, a common slope was used representing an average of all upper slopeswith an elasticity of negative one at the upper kink point. 29 Note that theKootenay Region was given a steeper slope than the common one since its upperkink point (at the base herd feed requirement level) was much closer to thevertical intercept. In other words, it was necessary to give the Kootenay Regiona greater slope, since the vertical intercepts to the upper slopes of the demandcurves should not necessarily exceed transportation costs between regions.Before the main results are presented, a typical year (simulation run) willbe described to show the flows that can occur in the allocation process. Thiswas done using the ‘Base Case Scenario’ parameters. In this typical year, therandomly drawn production levels for the regions are as follows: 143,857 tonsfor the Peace River Region; 262,373 for the Central Interior Region; 104,265 forthe Cariboo—Chilcotin Region; 334,676 for the Thompson—Okanagan Region; and,146,815 for the Kootenay Region. Note that the Central Interior, Cariboo—Chilcotin, and Kootenay regions’ production levels include quantities drawn fromstorage; these values equal 68,194, 15530, and 61,232 tons respectively. Giventhe regional base herd sizes, the regional base herd feed requirements are216,675 tons for the Peace River, 162,368 for the Central Interior, 208,734 forthe Cariboo—Chilcotin, 249,131 for the Thompson—Okanagan, and 63,461 for theKootenay regions. As a result, both the Peace River and Cariboo—Chilcotinregions did not produce enough forage to meet their requirements.With regions able to transport production within and between themselves,29 There was no concrete reason for deciding that an elasticity of negative one at the upperkink points was valid other than a general acceptance for assuming this by the Crop InsuranceBranch at British Columbia Ministry of Agriculture, Fisheries and Foods, and after consultationand debates with hay suppliers and consumers in B.C.41an equilibrium solution will occur when the Central Interior Region ships 72,818tons of forage to the Peace River Region and 27,187 tons to the Cariboo—ChilcotinRegion; the Thompson—Okanagan Region ships 77,282 tons to the Cariboo—ChilcotinRegion; and, the Kootenay Region ships 83,353 to storage. The result is that allregions are able to exactly satisfy there base herd feed requirement levels, withthe Thompson—Okanagan Region consuming 8,261 in excess of its base requirementlevel. Note that the differences in regional price levels in this equilibriumsolution do not exceed the costs of transportation, where the trading regionsprices exactly exceed transportation costs; the price in the regions equals 111dollars per ton (Peace River Region), 69 (Central Interior), 94 (Cariboo—Chilcotin), 72 (Thompson—Okanagan), and 41 (Kootenay) . The price in the KootenayRegion exactly exceeds the storage price 62 (expected price next year) by thecost of storage (due to spoilage)On average, the Peace River and Kootenay regions produce approximatelyenough forage to satisfy regional requirements. The Peace River Region produces195,333 tons of forage and requires tons 216,675, and the Kootenay Regionproduces 77,925 but requires only 63,461. The Cariboo—Chilcotin Region is asignificant net importer of forage, producing only 111,893 tons but requiring208,734. The Central Interior and Thompson—Okanagan regions are both netexporters, with the Central Interior producing 193,839 tons and consuming162,368, and the Thompson—Okanagan producing 318,272 and consuming 249,131.As a result of this mismatch in supply and demand for forage, crops willflow between and within regions. Note that storage is excluded from this exampleas steady—state for storage is assumed. The Peace River Region satisfies itsexcess demand by importing 21,342 tons of forage from the Central Interior.Cariboo—Chilcotin Region imports 10, 129 from the Central Interior and 83, 605 fromthe Thompson—Okanagan. Kootenay Region exports 14,464 to the Thompson—Okanagan.The result is that all regions more or less satisfy their forage requirements.Using the ‘Base Case Scenario’ parameters, 150 simulation runs werecreated. The results can be seen in Figures 5.1 to 5.5. The graphs show a plot421601401201100806040of each region’s predicted prices30 on the region’s supply levels. Alsoincluded are the average predicted price and production levels. Note that atypical simulation for the ‘Base Case Scenario’ was shown in the previousexample.° The predicted price values are similar to the actual prices. A description of the methodused for predicting prices will follow later.2003000000 50000 100000 150000 200000 250000Regional SupplyFigure 5.1 Peace River Region (Base Case Scenario)43Regionai SupplyFigure 5.3 Cariboo-Chilcotin Region (Base Case Scenario)I.mAveragePrice• •III.. ii• •S•AverageI0120100806040200140120100806040200S0 50000 100000 150000 200000 250000 300000Regional SupplyFigure 5.2 Central Interior Region (Base Case Scenario)350000SIAveragePriceAverageProduction0 50000-100000 20000015000044AveragePriceaIIIa.1120100801:20080706050a-3020100AverageProduction0 50000 100000 150000 200000 250000 300000 350000 400000 450000 500000Regional SupplyFigure 5.4 Thompson-Okanagan Region (Base Case Scenario)Is AverageIII Priceph I I IIIaIaa •aII Is IIIIaIIIa•aIaII0 20000 40000 60000 80000 100000 120000 140000 160000Regional SupplyFigure 5.5 Kootenay Region (Base Case Scenario)18000045From the graphs, predicted prices appear to rise as production levels fallas shown by the downward sloping trends. Since predicted prices are based on allregional production levels, and only one region’s production levels arerepresented in each graph, the observation do not exhibit a perfect trend (i.e.,the discrepancy in the trends of the observations can be attributed to the impactof other regions’ production levels on the price levels of the region inquestion)The predicted prices for the Peace River Region ranged from a low of 35dollars per ton to a high of 160, given an average of 108. The supply levels forthe Peace River Region from the 150 simulations ranged from 110,000 to 290,000tons. The Central Interior Region had predicted prices ranging from 4 to 118,given an average of 75, with supply ranging from 125,000 to 350,000. TheCariboo—Chilcotin Region had prices ranging from 25 to 135, given an average of96, with supply ranging from 70,000 to 150,000. The Thompson—Okanagan Region hadprices ranging from 8 to 120, given an average of 76, with supply ranging from180,000 to 475,000. The Kootenay Region had prices vary from 20 to 75, given anaverage of 48, with supply varying from 55,000 to 175,000.From the 150 simulation runs, regional prices and their associated quantitylevels were obtained. Regression analysis was used to draw a relationshipbetween these prices and quantities. Note that each regional price series wasregressed on all of the regional quantities using a linear OLS regressiontechnique. The regression equations are shown below.The results appear quite reasonable, with R—squared values ranging from0.59 for predicting price in the Peace River Region to 0.84 for the Cariboo—Chilcotin Region. The t—stat values (in brackets) also appear reasonable, withvalues greater than two for all but two cases. Given the significance of the t—stats, the Thompson—Okanagan Region appears to be one of the more significantregions affecting prices. This is not surprising since it is a considerablylarge exporter of forage, on the average. Note: ‘P’ refers to the region’sprice, ‘QPe’ to production in the Peace River Region; ‘QCe’ production in theCentral Interior; ‘QCa’ production in the Cariboo—Chilcotin; ‘QTh’ production in46the Thompson—Okanagan; and ‘QKo’ production in the Kootenay.Peace RiverP=322—0 . 0005*QPe_0 . 0002QCe—0 . 0002QCa0 . 0001kQTh0 .0000 9*QK0(14.6) (5.6) (3.0) (4.9) (2.4)R2=0 . 73Central InteriorP=294—0 . 0002*QPe_0 . 0003QCe0 . 0003QCa0 .0002 *QTh_0 . 000l*QK0(7.8) (13.0) (5.0) (10.6) (4.4)R2=0 .81Cariboo—ChilcotinP=309—0 . 000 1*QPe0 . 0003*QCe_0 . 0003QCa—0 . QQQ3*Q_ . 0002QKo(5.2) (11.2) (6.4) (16.0) (5.7)R2=0 .84Thompson—OkanaganP=291—0 . 0001*QPe_0 . 0002QCe—0 . 0003QCa0 . 0003QTh—0 . 0002QKo(2.5) (8.7) (6.1) (16.4) (5.2)R2=0 . 83KootenayP=150 .4—0. 00004*QPe_0 . 00009QCe—0 . 0001QCa—0 . 0002QTh0 . 0002QKo(1.3) (3.5) (1.9) (8.4) (6.5)R2=0 . 59Since a linear regression technique was used in creating the results above,an interpretation of the parameter values is relatively meaningless as actualvalue changes are represented. A more useful interpretation could be made if theparameter values represented percentage changes (or elasticities) . Therefore, theregressions were re—done using a log—linear form, since the associated parametervalues would represent percentage changes.The following equations show the results of the log—linear regressions usedin relating regional prices to all of the regional quantities. These are theequations that the Ministry of Agriculture would use in predicting regional pricelevels. The following variables are defined as%dQPE = the percentage deviation between actual and average production inthe Peace River Region%dQCE = the percentage deviation between actual and average production inthe Central Interior Region%dQCA = the percentage deviation between actual and average production inthe Cariboo—Chilcotin Region47%dQTH = the percentage deviation between actual and average production inthe Thompson—Okanagan Region%dQKO = the percentage deviation between actual and average production inthe Kootenay RegionAlso, let ‘%dPPE’ denote the percentage deviation between actual andaverage regional price in the Peace River Region; similar price deviationvariables are defined for the other regions.Log—Linear regressions for ‘Base Case Scenario’%dPPE=—1 . 05%dQPE—0 . 41%dQCE—0 . 25%dQCA—0 . 40%dQTH—0 . 1l%dQKQ%dPCE=—0 7l%dQPE—l 12%dQCE—0 46%dQCA—0.93%dQTH—0 22%dQKO%dPCA=—0 . 36%dQPE—0. 68%dQCE—0 . 37%dQCA—O. 96%dQTH—0. 17%dQKO%dPTH=--0 . 39%dQPE—0 . 72%dQCE—0 . 55%dQCA—1 . 36%dQTH—0. 24%dQKO%dPKO=—0 . 22%dQPE—0 . 41%dQCE—0 . 24%dQCA—1 . 00%dQTH—0. 40%dQKOThe previous equations can be interpreted as the effect that percentagechanges in all regional production has on the percentage change in each regionalprice. For example, if production was 10 percent below normal in all regions,then price would be 22.2 percent above normal in the Peace River Region. Thisvalue is calculated by multiplying —10% by the parameter associated with eachregion in the equation. Summing these values gives the cumulative effect of allregional production levels being 10 percent below normal on the price in thePeace River Region. Similarly, a 10 percent shortfall in production in allregions would lead to a price rise of 34.4 percent above normal in the CentralInterior, 25.4 percent above normal in the Cariboo—Chilcotin, 32.6 percent abovenormal in the Thompson—Okanagan, and 22.7 percent above normal in the Kootenayregions.As described at the end of Chapter 4, the slopes of the regional demandcurves was not known with certainty. The slopes of the demand curves, especiallythe upper ones play an integral role in determining the levels of the predictedprices, and this in turn can have an impact on the size and frequency of payoutsunder the WSD insurance scheme. It is for this reason that sensitivity analysiswas performed on the upper slopes of these regional demand curves. Thissensitivity analysis was done by increasing and decreasing the upper slopes of48the curves in the scenarios labelled ‘Steep Slope Scenario’ and ‘Flat SlopeScenario’, and re—doing the log—linear regressions to compare the effects onprice with that of the ‘Base Case Scenario’.The same log—linear regression procedure was used in the second scenario.In this case, the upper slopes of the regional demand curves were slightly higher(the vertical intercept on the upper slope was higher by 50 units from the ‘BaseCase Scenario’, for each region) . This scenario was labelled as the ‘Steep SlopeScenario’ . It was assumed that by increasing the slopes, higher price estimateswould be generated. See Appendix 2 for details.A comparison between the results from the ‘Base Case Scenario’ and the‘Steep Slope Scenario’ validates the assumption that increasing the slopes yieldslarger price estimates (i.e., steeper sloped demand curves are more priceresponsive). For the ‘Steep Slope Scenario’, the result of a 10 percentshortfall in production below normal yielded an increase in regional prices abovenormal by 25.1 for the Peace River, 40.0 for the Central Interior, 24.6 for theCariboo—Chilcotin, 33.8 for the Thompson—Okanagan, and 29.5 for the Kootenayregions. For all of these cases except for .he Cariboo—Chilcctin Region, thechange in price was greater than that under the ‘Base Case Scenario’. For thisregion, the ‘Steep Slope Scenario’ yielded a one percent lower increase in pricethan the ‘Base Case Scenario’, which could probably be attributed to aninaccuracy in parameterizing the regional demand curves, causing a slight biasin the allocation process (i.e., slightly more forage allocated to Cariboo—Chilcotin region when slopes are increased resulting in a less responsive price)This is, however, of no great concern as all other regions respond correctly andthe violation in the Cariboo—Chilcotin price is negligible in size.In the third scenario, the upper slopes of the regional demand curves wasdecreased by 50 units from the ‘Base Case Scenario’, for each region. It wasassumed that this would result in smaller price estimates (i.e., smaller slopeddemand curves are less price responsive) . See Appendix 2 for details.A comparison in results between the ‘Flat Slope’ and ‘Base Case’ scenariosconfirms the hypothesis of the lower slope being less price responsive. For the49‘Flat Slope Scenario’, a 10 percent shortfall in production below normal for allregions yields an increase in price above normal of 20.3 percent for the PeaceRiver, 29 percent for the Central Interior, 18.3 percent for the Cariboo—Chilcotin, 25.3 percent for the Thompson—Okanagan, and 14.2 for the Kootenayregions. For all of these cases, the increase in price was less than for the‘Base Case Scenario’The pricing model appears to respond reasonably well to changes in theslopes of the demand curves. Larger price estimates resulted from an increasein the slope in four of the five regions, confirming that steeper slopes are moreresponsive; and, lower price estimates for all regions resulted when slopes werelowered, showing that lower slopes are less price responsive. Next, the accuracyof the price estimates was checked.Using the linear regression results from the ‘Base Case Scenario’,predicted price estimates were obtained by substituting the regional quantitylevels from the 150 simulations back into each regression equation. A graphicaldisplay showing the accuracy of the price estimation is found below in Figures5.6 through 5.10. These graphs represent the predicted price over the actualprice for each of the 150 simulations. A value of one represents a perfectlyapproximated regional price.501.4IB . B1.2- B BI1B.B.•• B..B IB U B B1--I 1 B I B :•I••UII I • •B•B S08-— B BB0.2 -0-o 20 40 60 80 100 120 140Figure 5.6 Predicted over real price for Peace River Region2.5 -B20.a S1.5aI• •BS S Ba B BSal..5IhSSS1.I • •• •‘ I5•if a.. .a. • Id,Bdl..a.a Ba S• • IS0.5 - -B0- I I —I-- I I —0 20 40 60 80 100 120 140Figure 5.7 Predicted over re& price for Central Interior Region511.81.61.4 -$ a S&1.2--. • .a. • • ‘h. • • • •1 “ •. aI .? I.Q • a S a a.5 a00.40.20 I I —r I0 20 40 60 80 100 120 140 160Figure 5.8 Predicted over real price for Cariboo-Chilcotin Region1.61.4S a$1.2’ • aa •1.IS ••1.5•aa••5%SB a aa I aiIIiIa ••a. SB 1.• • a a..0.8-.0.4 -0.2- 1.0— I I I I0 20 40 60 80 100 120 140 160Figure 5.9 Predicted over real price for Thompson-Okanagan Region522.5 -I2-8I1.5UU U0 I I I0 20 40 60 80 100 120 140 160Figure 5.10 Predicted over real price for Kootenay RegionThe results appear quite reasonable, with all of the estimated regionalprices converging to their actual values (converging around the value one) . Onaverage for the regions, the model showed a rough accuracy of about 80 percent(varying between 1.2 and 0.8) . There were only a few extreme outliers found inthe four of the five regions over the 150 simulations, with Peace River being theexcluded region.The Central Interior Region exhibited two cases in which price wasunderestimated, with the more significant estimate being approximately 20 percentof actual price. Only two or three overestimates were found, with the largestbeing greater than two times the actual price. These extreme estimates are mostlikely a result of prices being generated from the kinked regional demand curves.Since prices along the inelastic portion of the demand curve are based on othertrading regions’ prices, and there is a significant jump from the upper to lowercurves, situations can possibly arise in which the price may appear overly high53or low. An example of this would be where a region produces excess productionand its surrounding regions have a significant excess demand. As a result theexcess supply will flow to the surrounding regions such that the region willconsume at its base requirement and have price generated from the other regionsprices. A jump in that region’s price level would be observed, as compared toif the regions had a strictly linear demand curves (no inelastic portion) . Sincethe regressions used are based on a linear fit, the results may appear to be anextreme over or underestimation in price.From a policy perspective, the implications of an overestimation in priceare more severe than an underestimation. If price is severely overestimated, andthe observation falls into the category in which a Wide—Spread Drought paymentmust be made (i.e., the estimated price exceeding the price trigger level and theactual supply falls below the guarantee level), then it can become quiteexpensive for the agency supplying the compensation.The Cariboo—Chilcotin Region only had one notable outlier in its priceestimates, where the estimated price exceeded the actual by approximately 60percent. The Thompson—Okanagan Region had one extreme underestimate, whereestimated price was approximately 20 percent of the actual and two overestimatesof approximately 50 percent over the actual. The Kootenay Region exhibited themost number of outliers. In over approximately five cases, the estimated priceexceeded the actual by more than 50 percent with the largest being more than 100percent. This can be explained by the fact that the Kootenay Region issignificantly isolated from the rest of the province, given the huge transportcosts especially beyond the Thompson—Okanagan Region. As a result the price inthe Kootenay Region may have relatively larger fluctuations in price than theother regions.3231 An observed overestimation may result if the region has excess supply like in the exampleabove, while underestimation may occur in the reverse if the region is in excess demand.32 Due to the large transport costs to the rest of the province the Kootenay Region doesalmost all of its trade with the Thompson-Okanagan Region; therefore, the Kootenay Region may54As stated previously, the purpose of this thesis was to develop a pricingmodel for forage. With this model, estimates of forage prices can be made andapplied directly to the problem of crop insurance. Forage prices are a necessarycomponent in a WSD insurance scheme since they are used to determine the levelof payments, and estimates are needed as prices are non—observable in the realworld. The way in which this model would be applied to the WSD insurance schemewill be described.Given the results from the regressions of prices on quantities (assumingthat the ‘Base Case Scenario’ parameters were used), the regression equations canbe used to obtain current price estimates. The Ministry of Agriculture (theinsurers) would substitute the current regional production levels for forage backinto each of the regression equations to obtain regional price estimates. Giventhese regional price estimates and their associated quantity levels, the insurerscould determine the indemnities (if any) owed to the farmers under the insurancescheme. As described previously, for a region to qualify for a payment underthis scheme, actual supply must fall below the guarantee level and price mustexceed a price trigger level.The following graph, Figure 5.11, provides a graphical description ofpayments under the WSD insurance scheme. The results from the 150 simulationsunder the ‘Base Case Scenario’ for the Peace River Region are shown in the plotsof predicted price on supply levels. Note that the price trigger andguarantee level34 are represented. Payment levels are equal to the quantitycoverage (amount below the guarantee level) multiplied by the price covered(amount above the price trigger) . The points in the top left quadrant representWide—Spread Drought payments, where supply falls below the guarantee level andthe estimated price exceeds the trigger level.have less price stability in extreme times with only one main trading path.This value was chosen to be 20% above the average price level.This value was chosen to be 80% of average production.551601401201100o 6040200The results from this graph are useful in that the insurers can determinethe frequency and cost of the program. In 150 simulations (years), the numberof times in which a payment is made to the Peace River Region can be ascertainedby the number of observations in the top left quadrant. The cost of thesepayments can be determined by the amount by which the observation exceeds theprice trigger and falls below the guarantee level. Note that both the frequencyand level of payments are of significant importance to the viability of theprogram.Both the frequency and levels will change as the price trigger and guarantee levels arechanged.0 50000 100000 150000 200000 250000 300000Regional SupplyFigure 5.11 Peace River Region (Base Case Scenario)56Chapter 6 : ConclusionsDuring times of wide—spread drought, forage producers and consumers, inparticular those in British Columbia, may experience drastic increases in forageprices. It had been proposed that a method be devised in which farmers couldinsure themselves against this price—risk associated with these types of naturaldisasters. The Wide—Spread Drought insurance scheme was proposed by the CropInsurance Branch at British Columbia Ministry of Agriculture, Fisheries, andFoods in order to deal with this issue. This insurance scheme did however,require the use of a model to estimate forage prices based on regional suppliesof forage, since prices were non—observable. The first portion of this thesisdeveloped a model to estimate forage prices. The second portion showed anapplication to British Columbia forage production, with a description of its usein a Wide—Spread Drought insurance scheme.Aside from the huge limitation on availability of data and vastcomplexities of the model, some reasonable results were obtained. Under the‘Base Case Scenario’ regression analysis was used to form a relationship betweenchanges in regional production levels and changes in regional price levels. Fromthe regressions, the R—squared values and t—statistics all appeared quitereasonable. The accuracy of the model in estimating prices was checked bygraphically representing predicted prices over real prices. With the exceptionof a few extreme values, the model worked quite well in price estimation.Results from two other sloped scenarios were compared with those of theBase Case in order to determine the models sensitivity to changes in the upperslopes of the regional demand curves. Log—linear regressions were performed sothat the changes in quantities and the effects on the price levels could beexpressed in percentages. It was assumed that increasing the slopes would makethe model more price sensitive and lowering the slopes, less price sensitive.The results validated this assumption as the ‘Steep Slope Scenario’ exhibited agreater change in regional price estimates above normal levels given anequivalent percentage shortfall in production below normal levels than the ‘Base57Case Scenario’. Note that this was true for all but the Cariboo—ChilcotinRegion, in which price was only one percent less responsive than in the ‘BaseCase Scenario’ . As well, the ‘Flat Slope Scenario’ exhibited a smaller changein regional price estimates above normal levels given an equivalent percentageshortfall in production below normal levels than the ‘Base Case Scenario’The model developed in the thesis shows considerable promise in itsapplicability to real world problems, in particular to the problem of cropinsurance. When applied to British Columbia forage production, it did reasonablywell in estimating forage prices and responded quite well to changes in theslopes of the regional demand curves. However, the accuracy of these results isquestionable as some of the necessary data was unavailable. In particular, therewas no forage consumption or price data available to parameterize the demandcurves. As a result, approximations to these curves had to be made. There wasno series data available on forage yields to obtain variance/covariance valuesfor the regions, therefore, these values were obtained by drawing relationshipsbetween variability in precipitation and variability in yields. There was alsolimited availability of data on average regional yield levels, amount of seededacreage used, and feed requirements per type of livestock.In order for the results to ultimately be useful for price estimationpurposes, all of the data requirements must be met. Without this completeinformation, any conclusions drawn regarding the results may be strictlyhypothetical.58BibliographyAgriculture Canada (1986), Sheep Production and Marketing, Ottawa.Ardeni, P.C. (1989), “Does the Law of One Price Really Hold for CommodityPrices?”, American Journal of Agricultural Economics, August, p.661—669.Aumack, Ken. 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(1976), “Field Loss Aspects”, Alfalfa Seminar, Kamloops, B.C.Statistics Canada (1987—1991), #22—002 Seasonal, November Estimates of PrincipalField Crops, Canada, Vol. 66—71, No. 8, Ottawa.Statistics Canada (1990), #22—201, Annual Statistics, Ottawa.Statistics Canada (1990), #22—201, Grain Trade of Canada, Ottawa.Statistics Canada (1991), #95—393D, Agricultural Profile of British Columbia1991 Census of Agriculture, Ottawa.Takayama, T., and W.C. Labys (1986), “Spatial Equilibrium Analysis”, Handbook ofRegional and Urban Economics, Vol. 1, p. 171—199.White, Kenneth J. (1993), Shazam : Econometrics Computer Program 7.0, User’sReference Manual. Montreal : McGraw—Hill, p.392.Willet, K. (1983), “Single— and Multi—Commodity Models of Spatial Equilibrium ina Linear Programming Framework”, Canadian Journal of AgriculturalEconomics, July, p.205—22.60Appendix 1OH()C)F1Q‘-CD0t(-DiOlDCD000H-11çtDi(lF-H-Oct()Cl)DJCD0b1111CD0LCDCD00H-CDCl)1-DO(-1-QOH‘<00H-IIIHI0CD IICD H-0 0HOC)‘-00CYDiCDCD00110DictH-çt()CDtbIiCD0Cl)0Dil)i00j—’J‘<0IHIC)I-<000CD‘sH-ct1DiHCD0C)11Di0H-LCD(-I-0DiH-10Ii-CbiCD—CDCDDJ11CDH’H’H’NJNJ-iCDLOC)H’C)C)C)U]C)CD LCD H-0 DiCD LCD H-H Di H CD‘CDctC)—H-ClflCD0•CDOCDOCDIjiH’H’11DiC)CDCD‘CDC)NJcc,,cc,C)a)cc,Cl)NJH-ctCDH)—OOH-’-’iOOCL‘CDOHCDOH’,),,)Di(i)CDCDD)C)C)U]CLCLH‘(-1-—H-i‘1t)CD0C)DiCDCDCD110.,CDCDcQU]C)Cl)C)r1-CDCLJF-’0H-XC)c,OCL0110U]C)10(0011NJa)C)Cl)’-<CLC))CDCD‘CDr1-r1-t’-’1tCD000CDCD10iQCDcQ(71UiU]a)WCl)CDCL—J-‘W-H-(I)C)a)C)C)tYCL—110C)—Jcc,U)(5)CDDiC)CDCD—J—1Uia)CD<•CLCD‘-0C)‘-0—C)HUjoDi Di H Di Cl) CD CD II CL ‘-‘1 CD CD CL CD CD CD H II CD CD DiCD CD Hh—-—H’C)U]U]DiU]cc,C)(iiC),NJNJU]—.i—-icc,NJU]0 II CDi_3)NJNJNJa)H’C)a)0—iU)Di H- CD Cl)CD ‘CD H-0 Di Di H H H CD Cl) 0 C) z CD CD 11 Cl)H Di H CD H’1$ CDCDCDDiCDNJNJH’NJDiQCDCl)cCDC)a)H’ItCDCLCDH-Hcc,C)NJa)H-00,.---11DictH’—iU]a)CDCDDiD)U]U]a)—1IHHHNJU]C)U]CLH’U]C)a)C)C)NJH’cc,H’C)C)cc,C)H’Cl)C)(I) Di CD CD‘CD H Di‘I‘-3C)C)C)00DiCDCD00Ii0CD(-I-H-ftC)CDt3IICl)0C)0CDCD00HCDL<0HC)H<00IiCD‘cH-ftt-CDHCD01)IICD0H-LCDft0CDH-Ii00CD H- 0 0OHCThC)HOC)0S3JPiOCDCD0DiDH-ftOl)JftiSHH-CDftC)Cr)CDC)0bhCD0LOrD00H-CDl)i00ct0OH<00H-I[1H-0 0I0CD 1(I)NJHUi.C)HCD0)HC)C)C)H-HHCDHHXHiHiQHC)C)ftpil)iCDC)HUiHUi0HF—QC)NJU)C)NJNJC)CDOOC)C) II CD CD LOHU)HHHO(C)C)C)C)U)OHGDNJSH0)—2C)Cl)0UiUiNJC)0—1HC)C)HftC)HCD0<CD<‘ICD 1 CDU)H0)HNJC)NJCDHC)0).C)0C)C)0CDNJNJU)Oo\011ft—1C-fl-0(DpiC)C)NJUiC)çtHiCDHCDC)C)U)NJ—1H-0LCDLCDC)C)NJ0-sCDCnH—-LCDCD0HH——0Cl)HCDHHOHQ0C)CDC)CDCDHC)C)C)C)C)bC)ciC)C)C)HO)HHCl)C)HC)HNJIH0O\0C)\OC)C)C)-\Oo\OI———C) ft H 0—3UiHHH0—’t2JO0—JHHC)C)(iiOHh)(D<‘C)HU)U)HC1CDOLCDCDC)-H-CDC)C1H-NJNJC)C)U)bHC)000iU)—1C)U)U)II00CC)NJU)C)U)DiNJftPiCDftC)H-HH-o\0000’l0Hi0—0h)-lCD LCD CDF-)H CD—F-<H)H(D OHsa0)C)H—3C)(0H C)C)HHH0Wl-<•.LQ.HC)C)‘IC)CD-1C)H—JNJo—-----lH----NJNJHHGDZ00C)H-C)C)C)U)C)rJC)(1)0U)C)—3UiCDCDHftfrC)UiC)C)0000C)C<CDCD—----H) 0 SH-CD H a C) CD N) C) ‘I CD F-h 0 N) 0 II CD LCD CD C) Fl 0 C) C)H CD CD-H CD (13HHC))C)C)C)ftC)(DOCCDN)ftQ_Q_C)C<CDH-C)HON)ftHlJ-<CD(DOHHaC) CD (I)C)NJOHflOl—fl000-0-CDOCDCD000H-hir10DictOHF-CDrr0Cl)CDtI00-hlhiCD00CD00H-CDCD00çtQOH<00F-IhiF CD IICD F 0 071OHflO—10O0‘c0-0-CDOCDCD000H-hiitOCDit:IHF-CDrh0CDCDO00-hihiCD0CPu)00H-CDCD00itOOHk<00F-IhiFI0CD hiCD H 0 00C-) 0- 0 H CD (I) wi-rnOlDN)HCO01CDri-‘--hihiCD0]COC)F-CD01CDCO01OH01—JCDohiH CD 0- H CD CO10 CD 0 0 0 C’) CD a CD i-I- hi H H 0 hi hi 0 a 0 C) it F- 0 0:i’-oH-CDCDCDCDCDCD<CD.CDON)010(i1hiCDCOCO0]NCO0101CO0]CO—Jl—CD.)h-rnOCDPPPPD0101CO.O01hihia-CN)N)01F-CDCO01CO0]01HCOCO—1CDCDhi on0-CDPPPPP1-CO01CO00-01N)i-N)0100—JCDCOCOCO003—101—3l-OHs0-PPPPPCONCO0103CDd01CON)N)CON)—1CD010)00COCO—ICOhi CDC)t,0 JH1 H]çt H-0 00 it 0 H] CD hi H CD it F- 0 0 C) 0 hi hi CD H CD it H- 0 0 C-) 0 CD Nh Nh H C) H- CD ID it CD it hi F Nh 0 hi Ii’ CD H- 0 ID CD Hon0-CDf-H-hih-COHH--‘00-—CDCD00C_flCDC)ItoCO01F-IID OHs0-0001ID CDt5C)CDCDCDCCII’)000100H CD 0- H CD 01CD N)CO 03CDCDCDCD0 0 it CD ID CD 1<I’l 0CDCDCDCDCD0..itNCO01N)CD010103COCOIDN)N)-JCOC’)CDN)CO01CO—10 0 CI-CD Di 1<CD Di C) CD H CD IIH 0 Di 0 C) Di C) Di Di C)0 Di II H 0 0 C H- H C) 0 CI- H- C)0 CD C) CI- Ii Di H F-I C) Cr CD II H- 0 IICD H 0 C)OF-)00HO00C)DiC)CDCl)0DiOH-HCrC)DiCrC)HH-CDrrC)Cl)DiZi0bIF-ICDC)ODi00H-DiDiDiOtoOHJ<C)C)H-IIIHIC)CD IICD H-0 C)CDtj* F-I H- 0<(DODill Di CC) CD II H- C) CD 0 Ct H- C) CD a Mi H 0 Di C) cli CD Di CD 0 Hi CI C)- CD H CD H 0 C) Di H CD 0 Fl CI- Di C) a H 0 11 c-I-(n<zIDiOHH-0HCDCDC)C)0CD1L‘)(DO(iiI-il0101(51c-Io\°o\OO\OO\OO\OcziC) CD H 0I-C)0<HODiH0Di(DCC) CDI-CC)0)C)0)—JNJCo(ii01t’)H-CD0CC)CDH-*0 C) Di HCD CQ H 0 C) Di H C,)CI-0 II Di CO CD 0 0 Di CtDi a 0 H H Di II Di CD II CC 0 C)H Di H CD C)CJi-C)DiC)C)C1•CDHDiH-UiH—•CD DiAH 001-Cl)H0cD HH-CD01H—010IDDiH--Cl)HC)CD.DiH 01 01-Cl)IDCIHC)NJH-•0 Diwa010IDa H010C)H-UiHIDCD DiI H C)H H-H CD DiC)H H V———-U H-H-CD<Di00—JC)aC,I-lCDHC)HC)NJH H Di C) DiHO0C)CDHc-I-C)Cr(DCI-DiC)-NJHCl-FIHHUi01Ui[)0H0 0 Di——Cr DiC)DiH-HHH-DiC)]NJHNJC)NJUi01‘Cr0H-H-IC)a 0—-—HOHHDcC)DiOC)HNJDiI-tINJ01NJ01HCDC)C)II 0 C)H Di H CD -UNJHHNJHC)HC)OCDDiCQCtH 0 C) Di H (I) c-C 0 F-I Di CO CDH01C)—J01NJC)HC)0 0 CD C) Di 1<65Appendix 266Sensitivity AnalysisLog—Linear regressions for ‘Steep Slope Scenario’%dPPE=—1 22%dQPE—O . 44%dQCE—O . 33%dQCA—O. 44%dQTH—O 08%dQKO%dPCE=—O 77%dQPE—1 17%dQCE—O 58%dQCA—]. 18%dQTH—O 30%dQKO%dPCA=—O 39%dQPE—O 58%dQCE—O . 34%dQCA—O . 96%dQTH—D . 19%dQKO%dPTH=—O 55%dQPE—O 74%dQCE—O 52%dQCA—1 34%dQTH—O 23%dQKO%dPKO=—O 45%dQPE—O . 35%dQCE—O. 47%dQCA—1 06%dQTH—O. 62%dQKOLog—Linear regressions for ‘Flat Slope Scenario’%dPPE—1 05%dQPE—O 41%dQCE—O . 24%dQCA—O 30%dQTH—O 03%dQKO%dPCE=—O. 61%dQPE—O. 99%dQCE—O . 39%dQCA—O . 76%dQTH—O. 15%dQKO%dPCA=—O .2 6%dQPE—O . 50%dQCE—O . 27%dQCA—O. 68%dQTH—O. 12%dQKO%dPTH=—O . 31%dQPE—O 63%dQCE—O . 34%dQCA—1 14%dQTH—O. 11%dQKO%dPKO=—O . 1O%dQPE—O 32%dQCE—O . 12%dQCA—O . 53%dQTH—O 35%dQKO67Appendix 3* GAMS file to run the spatial equilibrium model* DEFINING THE FILE TO WRITE TOFILE outt /outt.dat/;FILE con / con.dat/;FILE tostor /tostor.dat/;FILE storm /storin.dat/;* SET(S)” DECLARES THE SETS (DOMAIN) OVER WHICH THE PARAMETERS* AND VARIABLES ARE DEFINED (I.E. VALUES FOR EACH REGION). PAGES* 5 TO 7 OF A GUIDE TO USING GAMS.SETR REGIONS/PEACECENTRALCAR IBOOTHOMPSKOOTENSTORAGE!;SETU ITERATIONS/Ul*U150/;SETY REPETITIONS FOR SOLVING PRICE/Y1*Y7/;* PARAMETER (FOR ENTERING PARAMETER LISTS, I.E. VECTORS)* TABLE (FOR TWODIMENSIONAL TABLES, I.E. MATRICES)* SCALAR (FOR SCALARS, I.E. SINGLE ELEMENTS, CONSTANTS)PARAMETER ALU(R) UPPER INTERCEPT OF DEMAND CURVES/PEACE 225CENTRAL 154CARIBOO 194THOMPS 239KOOTEN 239STORAGE 78/;PARAMETER ALD(R) LOWER INTERCEPT OF DEMAND CURVES!PEACE 251CENTRAL 180CARIBOO 220THOMPS 265KOOTEN 91STORAGE 0/;PARAMETER BU(R) UPPER SLOPE OF DEMAND CURVES/PEACE 0.00052CENTRAL 0.00044CARIBOO 0.00051THOMPS 0.00055KOOTEN 0.0027STORAGE 0.000189/;PARAMETER BL (R) LOWER SLOPE OF DEMAND CURVES/PEACE 0.00075CENTRAL 0.00075CARIBOO 0.00075THOMPS 0.00075KOOTEN 0.00075STORAGE 0/;* THE PARAMETER CEAR REPRESENTS THE CONSUMPTION REQUIREMENT, le THE* VERTICAL PORTION OF THE DEMAND CURVE. IT IS REPRESENTED BY NUMBER OF* COWS MULTIPLIED BY THE FEED REQUIREMENT PER COW.PARAMETER CB(R) CONSUMPTION REQUIREMENT/PEACE 216675CENTRAL 162368CARIBOO 208734THOMPS 249131KOOTEN 63461STORAGE 413136/;*NOTE BELOW THAT THE MEAN VALUES FOR THESE INDEPENDENT RANDOM QUANTITIES IS*ACCOUMTED FOR BY ADDING A VECTOR OF MEAN VALUES M TO THE VECTOR OR*CORRELATED QUANTITIES, OBTAINED BY MULTIPLYING BY THE CHOLESKY MATRIX.*THE VARIANCES FOR THE INDEP’s IS ACCOUNTED FOR IN THE CHOLESKY*DECOMPOSITION MATRIX (i.e. *FROM THE COVARIANCE MATRIX)PARAMETER IND(R) INDEPENDENT RANDOM QUANTITY DRAWS;IND ( ‘PEACE’ ) =NORMAL (0, 1);IND( ‘CENTRAL’ )=NORMAL(0, 1);IND ( ‘CAR IBOO’ ) =NORMAL (0, 1);IND ( ‘THOMPS’ ) =NORMAL (0,1);IND ( ‘KOOTEN’ ) =NORMAL (0,1);PARAMETER M(R) MEAN QUANTITIES PRODUCED/PEACE 195333CENTRAL 193839CARIBOO 111893THOMPS 318272KOOTEN 77925/;ALIAS (R,RP);*R IS ROWS AND RP IS COLUMNSTABLE CHOL(R,RP) CHOLESKY DECOMPOSITION MATRIX FOR THE RANDOM QUANTITIESPEACE CENTRAL CARIBOO THOMPS KOOTENPEACE 34539 0 0 0 0CENTRAL 4136 24956 0 0 0CARIBOO 2383 3980 16706 0 0THOMPS 5078 6607 6703 45332 0KOOTEN 938 1033 1059 1715 10283;PARAMETER QP(R) CORRELATED RANDOM QUANTITIES EXCLUDING CURRENT STORAGE;QP(’ PEACE’ )=SUM(RP,CHOL( ‘PEACE’ ,RP) *IND(RP) ) +M( ‘PEACE’);QP(CARIBOO)=SUM(RP,CHOL(CARIBOOi,RP)*IND(RP))+M(ICARIBOO);QP ( ‘THOMPS’ ) =SUN(RP, CHOL ( ‘THOMPS’ , RP) *IND(Rp) ) +M( ‘THOMPS’);QP( ‘PEACE’) =MAX(QP( ‘PEACE’) , 0);QP(’CENTRAL’)=MAX(QP(’CENTRAL’),O);QP ( ‘CARIBOO’ ) =MAX (QP ( ‘CARIBOO’ ) , 0);QP(’THOMPS’ )=MAX(QP(’THOMPS’) , 0);QP ( ‘KOOTEN’ ) =MAX (QP ( ‘KOOTEN’) , 0);PARAMETER PRICE(R) PRICE VALUES CALCULATED INSIDE THE LOOP;PARAMETER S1(R,U) CURRENT STOR ADDED TO CURRENT PROD IN RUN2 FROM SERIES 1;PARAMETER S2(R,U) CURRENT STOR ADDED TO CURRENT PROD IN RUN3 FROM SERIES 2;7OPARAMETER S3 (R, U) CURRENT STOR ADDED TO CURRENT PROD IN RTJN4 FROM SERIES 3;PARAMETER S4(R,U) CURRENT STOR ADDED TO CURRENT PROD IN RUN5 FROM SERIES 4;PARAMETER S5(R,U) CURRENT STOR ADDED TO CURRENT PROD IN RUN6 FROM SERIES 5;* NORMALLY TABLES ARE DEFINED OVER TWO DIFFERENT SETS. EXAMPLE,* IN THE STANDARD LP PROBLEM, THE TWO SETS ARE ACTIVITIES* (COLUMNS) AND INPUTS (ROWS) OF THE COEFFICIENT MATRIX. IN THE* PRESENT 3 AGENT PROBLEM, THE R SET IS USED TO DEFINE BOTH THE* COLUMNS AND THE ROWS OF THE TRANSPORTATION COST MATRIX. NOTE,* WE HAVE TO USE THE ALIAS COM[vIAND (P.35) TO GIVE THE R SET* ANOTHER NAME. WE WILL CALL THE NEW NAME RP (I.E. R ‘PRIME’).* THE FIRST SCRIPT IN THE TABLE REFERS TO THE ROWS AND THE* SECOND REFERS TO THE COLUMNS (P.26).TABLE A(R,RP) TRANSPORTATION COSTS BETWEEN REGIONSPEACE CENTRAL CARIBOO THOMPS KOOTEN STORAGEPEACE 7 42 48 61 78 18CENTRAL 42 13 25 43 61 21CARIBOO 48 25 13 22 56 16THOMPS 61 43 22 13 42 22KOOTEN 78 61 56 42 13 21STORAGE 18 21 16 22 21 0;* VARIABLES CAN BE EITHER POSITIVE, NEGATIVE, INTEGER, BINARY, OR* FREE (CAN TAKE ON ANY VALUE). GAMS DEFAULTS TO FREE VARIABLES.VARIABLES0(R) CONSUMPTION IN EACH REGION (AGENT)TR(R,RP) AMOUNT TRANSPORTED BETWEEN REGIONSW TOTAL WELFAREIA(R) INDICATOR FOR C LESS THAN CBAR13(R) INDICATOR FOR C GREATER THAN CBARPOSITIVE VARIABLES C,TR,IA,IB;* EQUATIONS NEED TO FIRST BE DESCRIBED (DECLARED) AND THEN* DEFINED. WHEN DEFINING EQUATIONS, THEIR NAMES ARE FOLLOWED BY* TWO DOTS. =E= DENOTED EQUAL TO, =G= GREATER THAN EQUAL TO, AND*=L= LESS THAN EQUAL TO.EQUATIONSWELFARE OBJECTIVE FUNCTIONCONSUM(R) CONSUMPTION CONSTRAINTSQUANT(R) QUANTITY CONSTRAINTSSTOREA(R) STORAGEA CONSTRAINTSTOREB(R) STORAGEB CONSTRAINTICONA(R) ICONSTRA CONSTRAINTICONB(R) ICONSTRB CONSTRAINTICONC(R) ICONSTRC CONSTRAINTICOND(R) ICONSTRD CONSTRAINTICONE(R) ICONSTRE CONSTRAINT;WELFARE.. W=E=STJN(R,IA(R)*(ALU(R)*C(R)_0.5*BU(R)*C(R)*C(R))÷IB(R)*(ALU(R)*CB(R)_0.5*BU(R)*CB(R)*CB(R)+ (0(R) —CB (R) ) * (ALD(R) —BL (R) * (0(R) +CB (R) ) /2) )-SUM( (R,RP) ,A(R,RP) *T(,p));CONSUM(R).. OCR) =L=SUMCRP,TRCR,RP));QUANTCR).. QP(R)=G=SUM(RP,TRCRP,R));STOREACR).. TR(’STORAGE’,R)=G=O;STOREBCR).. TR(R, ‘STORAGE’)=E=O;IA(R) * (C (R) -CB(R) ) =L=O;13CR) * CC CR) -CB CR) ) =G=O;IA CR) ÷13 CR) =E=1;IA CR) =L=1;lB CR) =L=1;*.L DESIGNATES A STARTING VALUE AND .FX DENOTES A FIXED* VARIABLE, CP. 47 TO 48). WE WILL USE THE AUTARKY SOLUTION AS OUR* STARTING VALUES (NO TRADE). THE DEFAULT STARTING VALUE IS 0.C.LCR) = QP(R);*“MODEL’ IS USED TO NAME THE MODEL AND TO IDENTIFY THE EQUATIONS* WHICH IT INCUDES, (P.11 TO 12). WE CALL OUR MODEL ‘VERSIONi’ AND* TELL GAMS TO INCLUDE ALL OF THE EQUATIONS, I.E. WELFARE,* CONSUMPTION AND QUANTITY PRODUCED.MODEL VERSION1 /ALL/;* HERE IS A TITLE FOR THE PRICES AND QUANTITES SENT TO outt.cIatPUT outtPUT “ RESULTS FROM GAMS OUTPUT!PUT“PUT “ QP1 P1 QP2 P2 QP3 P3 QP4 P4 QP5 P5/;PUT conPUT “ CONSUMPTION” /PUT “ “1/PUT “ CONPEA CONCEN CONCAR CONTHO CONKOO “I;PUT tostorPUT “ ALLOCATION TO STORAGE’!PUT “ “1/PUT “ PRICESTO STOPEA STOCEN STOCAR STOTHO STOKOO “1;storm“STORAGE ADDED TO CURRENT PRODUCTION”!II / /STOPEA STOCENOPTION SEED = 2576;*CREATE THE LOOPS TO GENERATE THE PRICE DISTRIBUTION OVER SET U*********5TART OF FIRST LOOPLOOP CU,IND C ‘PEACE’ ) =NOFMAL C 0, 1);IND C ‘CENTRAL’ ) =NORMAL (0, 1);IND C ‘CARIBOO’ ) =NORMAL C 0, 1);IND C ‘THOMPS’ ) =NORMAL CD, 1);IND C ‘KOOTEN’ ) =NORMAL (0, 1);QP( ‘PEACE’ )=SUM(RP,CHOL( ‘PEACE’ ,RP) *INDCRp) )+MC’PEACE’);QPC ‘CARIBOO’ )=SUN(RP,CHOL( ‘CARIBOO’ ,RP) *INDCRp) ) +MC ‘CARIBOO’);QPC ‘THOMPS’ ) =SUMCRP,CHOL C ‘THOMPS’ ,RP) *INDCRP) ) ÷MC ‘THOMPS’);QP C ‘KOOTEN’ ) =SUMCRP, CHOL C’ KOOTEN’ ,RP) *IND CRP) ) ÷MC ‘KOOTEN’);QP C ‘PEACE’ ) =MAX (QP C ‘PEACE’) , 0);QP(’CENTRAL’ )=MAX(QPC’CENTRAL’) , 0);QP(’CARIBOO’)=MAX(QPC’CARIBOO’ ) ,0);QP C ‘THOMPS’ ) =MAXCQP C ‘THOMPS’) , 0);QP C ‘KOOTEN’ ) =MAX(QP C’ KOOTEN’) , 0);ICONA(R)ICONB CR)..ICONC(R)..ICONDCR)..ICONE(R)..7,PUTPUTPUTPUT STOCAR STOTHO STOKOO “/;C.L(R)=QP(R);72.* SOLVE INDICATES : (A) THE MODEL TO BE SOLVED; (B) THE* DIRECTION OF SOLUTION (MAX/MIN); (C) THE NAME OF THE OBJECTIVE* VARIABLE; (D) THE SOLUTION PROCEDURE TO BE USED.SOLVE VERSION1 MAXIMIZING W USING NLP;Si ( ‘PEACE’ , U) = TR. L ( ‘STORAGE’, ‘PEACE’);Si ( ‘CENTRAL’ ,U) = TR. L (‘STORAGE’, ‘CENTRAL’);Si ( ‘CARIBOO’ ,U) = TR.L( ‘STORAGE’, ‘CARIBOO’);Si ( ‘THOMPS’ ,U) = TR. L ( ‘STORAGE’ , ‘THOMPS’);Si ( ‘KOOTEN’ ,U) = TR. L (‘STORAGE’, ‘KOOTEN’);********END OF FIRST LOOPOPTION SEED = 3367;*********5TART OF LOOP 2LOOP (U,IND ( ‘PEACE’ ) =NORMAL (0, 1);IND ( ‘CENTRAL’ ) =NORMAL (0, 1);IND ( ‘CARIBOO’ ) =NORMAL (0, 1);IND (‘THOMPS’ ) =NORMAL (0, 1);IND(’KOOTEN’)=NORMAL(O,l);QP ( ‘PEACE’ ) =SUM(RP, CHOL (‘PEACE’ ,RP) *IND(Rp) ) +M( ‘PEACE’ ) +S1 (‘PEACE’ , U);QP(’CENTRAL’)=StJM(RP,CHOL(’CENTRAL’,RP)*IND(RP))+M(CENTRAL)+S1(CENTRAL,U);QP( ‘CARIBOO’ ) =SUM(RP,CHOL( ‘CARIBOO’ ,RP) *IND(RP) ) +M( ‘CARIBOO’ ) +S1 ( ‘CARIBOO’ ,U);QP ( ‘KOOTEN’ )=SUN(RP,CHOL( ‘KOOTEN’ ,RP) *IND(Rp) ) +M( ‘KOOTEN’ ) +S1 ( ‘KOOTEN’ ,U);QP ( ‘PEACE’ ) =MAX (QP ( ‘PEACE’) , 0);QP(’CENTRAL’)=MAX(QP(’CENTRAL’) ,0);QP ( ‘CARIBOO’ ) =MAX (QP ( ‘CARIBOO’) ,0);QP ( ‘THOMPS’ ) =MAX(QP ( ‘THOMPS’) ,0);QP( ‘KOOTEN’ )=MAX(QP(’KOOTEN’) , 0);C. L (R) =QP (R);SOLVE VERSION1 MAXIMIZING W USING NLP;S2 ( ‘PEACE’ ,U) =TR. L ( ‘STORAGE’, ‘PEACE’);S2(’CENTRAL’ ,U)=TR.L(’STORAGE’,’CENTRAL’);S2(’CARIBOO’ ,U)=TR.L(’STORAGE’, ‘CARIBOO’);S2 (‘THOMPS’ ,U) =TR.L( ‘STORAGE’, ‘THOMPS’);S2 ( ‘KOOTEN’ ,U) =TR.L (‘STORAGE’, ‘KOOTEN’);*********END LOOP 2OPTION SEED =2107******5TART LOOP 3LOOP (U,IND(’PEACE’ )=NORMAL(0,1);IND(’CENTRAL’)=NORMAL(O,l);IND ( ‘CARIBOO’ ) =NORMAL (0, 1);IND ( ‘THOMPS’ ) =NORMAL (0, 1);IND(’KOOTEN’)=NORMAL(O,l);QP(’PEACE’)=SUM(RP,CHOL(’PEACE’ ,RP) *IND(RP) )+M(’PEACE’ )+S2(’PEACE’ ,U);73QP ( ‘CENTRAL’ ) tSUN (RP, CHOL (‘CENTRAL’ ,RP) *IND (RP) ) +M( ‘CENTRAL’) +S2 (‘CENTRAL’ ,U);QP(’KOOTEN’)SUM(RP,CHOL(’KOOTEN’,RP)*IND(RP))÷M(KOOTEN)+S2(KOOTEN,U);QP ( ‘PEACE’ ) =MAX (QP ( ‘PEACE’) , 0);QP ( ‘CENTRAL’ ) =MAX (QP ( ‘CENTRAL’) ,0);QP ( ‘CARIBOO’ ) =MAX (QP ( ‘CARIBOO’) ,0);QP (‘THUMPS’ ) =MAX(QP ( ‘THOMPS’) ,0);QP( ‘KOOTEN’ ) =MAX(QP ( ‘KOOTEN’) ,0);C.L(R)=QP(R);SOLVE VERSION1 MAXIMIZING W USING NLP;S3 (‘PEACE’ ,U) =TR. L ( ‘STORAGE’, ‘PEACE’);53 (‘CENTRAL’ ,U)=TR.L(’STORAGE’,’CENTRAL’);S3(’CARIBOO’,U)=TR.L(’STORAGE’,’CARIBOO’);S3 (‘THUMPS’ ,U) =TR. L ( ‘STORAGE’, ‘THOMPS’);S3 (‘KOOTEN’ ,U) TR.L (‘STORAGE’, ‘KOOTEN’);******END LOOP 3OPTION SEED = 2688;*********5TART LOOP 4LOOP (U,IND ( ‘PEACE’ ) =NORNAL (0, 1);IND ( ‘CENTRAL’ ) =NORMAL (0, 1);IND ( ‘CARIBOO’ ) =NORNAL (0, 1);IND(’THOMPS’)=NORMAL(O,l);IND(’KOOTEN’)=NORMAL(O,l);QP ( ‘PEACE’ ) =SUN(RP, CHOL (‘PEACE’ ,RP) *IND (RP) ) +M( ‘PEACE’ ) ÷S3 (‘PEACE’ ,U);QP (‘CENTRAL’) =STJN (RP,CHOL (‘CENTRAL’ ,RP) *IND(RP) ) +M( ‘CENTRAL’) ÷S3 (‘CENTRAL’ ,U);QP( ‘THOMPS’ ) =SUM(RP,CHOL( ‘THOMPS’ ,RP) *IND (RP) ) +M( ‘THOMPS’ ) ÷S3 (‘THOMPS’ ,U);QP ( ‘KOOTEN’ ) =SUN(RP, CHOL ( ‘KOOTEN’ ,RP) *IND (RP) ) ÷M( ‘KOOTEN’ ) ÷S3 (‘KOOTEN’ ,U);QP ( ‘PEACE’ ) =MAX (QP ( ‘PEACE’) ,0);QP( ‘CENTRAL’) =MAX (QP( ‘CENTRAL’) , 0);QP(’CARIBOO’)=MAX(QP(’CARIBOO’),O);QP(’THOMPS’)=MAX(QP(’THOMPS’),O);QP(’KOOTEN’)=MAX(QP(’KOOTEN’),O);C.L(R)=QP(R);SOLVE VERSION1 MAXIMIZING W USING NLP;S4 (‘PEACE’ , U) =TR.L( ‘STORAGE’ , ‘PEACE’);S4 (‘CENTRAL’ ,U) =TR. L (‘STORAGE’, ‘CENTRAL’);84 ( ‘CARIBOO’ ,U) =TR. L( ‘STORAGE’, ‘CARIBOO’);S4 ( ‘THOMPS’ ,U)=TR.L( ‘STORAGE’, ‘THOMPS’);S4 ( ‘KOOTEN’ ,U) =TR.L (‘STORAGE’, ‘KOOTEN’);********END LOOP 4OPTION SEED = 4444;*******START LOOP 5LOOP (U,IND ( ‘PEACE’ ) =NORNAL (0, 1);IND ( ‘CENTRAL’ ) =NORMAL (0, 1);IND ( ‘CARIBOO’ ) =NORMAL (0, 1);IND ( ‘THOMPS’ ) =NORMAL (0, 1);IND(’KOOTEN’)=NORMAL(0,1);QP( ‘PEACE’ )=SUM(RP,CHOL(’PEACE’ ,RP) *IND(RP) )+M(’PEACE’ )+S4 (‘PEACE’ ,U);QP ( ‘THOMPS’ ) =STJN(RP,CHOL( ‘THOMPS’ ,RP) *IND(Rp) ) +M( ‘THOMPS’ ) +S4 (‘THOMPS’ ,U);QP ( ‘KOOTEN’ ) =SUN(RP, CHOL ( ‘KOOTEN’ ,RP) *IND (RP) ) +M( ‘KOOTEN’ ) +S4 (‘KOOTEN’ ,U);QP(’PEACE’)=MAX(QP(’PEACE’),O);QP(’CENTRAL’)=MAX(QP(’CENTRAL’) ,0);QP(’CARIBOO’)=MAX(QP(’CARIBOO’),O);QP ( ‘THOMPS’ ) =MAX(QP ( ‘THOMPS’) ,0);QP ( ‘KOOTEN’ ) =MAX(QP ( ‘KOOTEN’) ,0);C.L(R)=QP(R);SOLVE VERSION1 MAXIMIZING W USING NLP;S5 ( ‘PEACE’ ,U) =TR - L ( ‘STORAGE’, ‘PEACE’);S5 (‘CENTRAL’ ,U)=TR.L( ‘STORAGE’ ,‘CENTRAL’);S5 (‘CARIBOO’ ,U)=TR. L( ‘STORAGE’ , ‘CARIBOO’);S5 ( ‘THOMPS’ ,U) =TR.L (‘STORAGE’, ‘THOMPS’);S5 (‘KOOTEN’ ,U) =TR.L (‘STORAGE’, ‘KOOTEN’);********END LOOP 5OPTION SEED = 4111;********5TT LOOP 6LOOP (U,IND(’PEACE’)=NORNAL(O,l);IND ( ‘CENTRAL’ ) =NORNAL (0, 1);IND ( ‘CARIBOO’ ) =NORNAL (0,1);IND ( ‘THOMPS’ ) =NORMAL (0, 1);IND(’KOOTEN’)=NORMAL(O,l);QP( ‘PEACE’ )=SUN(RP,CHOL(’PEACE’ ,RP) *IND(Rp) )÷M(’PEACE’ ) ÷S5 (‘PEACE’ ,U);QP ( ‘CENTRAL’ ) =SUN (RP, CHOL ( ‘CENTRAL’ ,RP) *IND (RP) ) ÷M( ‘CENTRAL’ ) +S5 ( ‘CENTRAL’ ,U);QP ( ‘THOMPS’ ) =SUM(RP, CHOL ( ‘THOMPS’ ,RP) *IND (RP) ) +M( ‘THOMPS’ ) ÷S5 ( ‘THOMPS’ ,U);QP ( ‘KOOTEN’ ) =SUN(RP, CHOL ( ‘KOOTEN’ ,RP) *IND (RP) ) +M( ‘KOOTEN’ ) ÷S5 ( ‘KOOTEN’ ,U);QP ( ‘PEACE’ ) =MAX (QP ( ‘PEACE’) ,0);QP( ‘CENTRAL’) =MAX (QP( ‘CENTRAL’) , 0);QP( ‘CARIBOO’ ) =MAX(QP( ‘CARIBOO’ ) ,0);QP(’THOMPS’)=MAX(QP(’THOMPS’),O);QP ( ‘KOOTEN’ ) =MAX(QP ( ‘KOOTEN’) ,0);C.L(R)=QP(R);SOLVE VERSION1 MAXIMIZING W USING NLP;* SOLUTION REPORT* GENERATING THE PRICESC.L(R)=ROUND(CL(R),4);cicici(Nfl(NH”ciH•CN—H124124ci(N—riO”H0—•O(N0-r’ihIcNH—hflZUUHH”0Hh’IHH-I]‘—00ciH0III124124r(3(J12)Hc1(JQ•i:I111HzIXhilOUHUOU-111-UU000LI)UCD1’)U12—(l)-HH-124•1110C!)CDH--ci01U-mci124H011413-i”“•ccicici(N(N--Cl)II124111H”••“HHci•o•..H••—••HH(:10“O——-)i-)H—0(00H•-13)134Hci13Cl)1-r1H.hi1.h1(I)•ci111111HH1IU0IXHHUHOHOH(NHh’IUHOO13.--riH:hi1U-U-——12.hrl----S---HI121121121111010—inU)UUU-1313UH.--1•LflCl)cifl.Hci130Cl)“HHH01Cr..13-ciUci0-1111U—U---Cl)(3.0’i•.cici..H11lhi)••ciQci..•.C101101+Hci—•••.c’C’H—--H.“C’—(NHH..124134H(Ni-)H-H..•.-.11)”i-li-li-I—i—13HH”Z..——D0111111Z••13-Ii)—--00cDUUU‘l—l’l—H-H-Cl)--i-)U1U-ZCl)O111124-Cl)O1312134•HH011111111240U111124HO00UUUUOU(l)HH01-il-0-fi)h..icD•--(0(l)—J}---13.H11It-1111’l(0I3.—II—IIH“-fl.1’IHhI--•d13.H01’I—---1’.UUU-J-U-U•-------s-i--4JU)-)0HHH...—----._-----0—H-H-)i-)0—--’tflH-12—-13h±101110ZZ12.12.l30•JLfl01313>UUH00113401134OUU(i)(l)(l)-J13.HHZ111HHHHHHHHO00124131-TIDDDDDtDDDO00Cl)13.13.12413413413.124134F-Il-)****(0(0(0(*******t0t***(00****(I)i-hi-h****CDCDCDCDCDCDCDCDCD(DCDCDCDCDCDCDCD(DCD(D(DCDCDCDCDCDCDCDCDCDCDCDCDCDCD(D(DCDDCDH-H--M.Cl)‘IjCDHHCD00OCDMFH-H-i-DiP1’tiCDCD1•CD<0i-(D—IIl-—Hl)(l)CfltflCflfLj)Cl)tl)CDH-<’t50H-CDCDaCDCDIOH-0000000000LI•QCDLI1WD-JHhQtsJH—ft(DMWHU1HHWPOU1iiIiWilt-JilHilH-H-HiI01111111111O(DIIIH00H-IIIIIIIIIIIIIIIIIIIIIiIIIIIIIIIIIIIIIIWWDJ’J’)HHHH0Dl000H-00WUiW’J0(Ql-N)HHPHH—i-h••HHHOOOD)OD.JOHOHOHOHOU1*-*W*J*H*Li-OilHIIIIIIIIIIIIIiCDCD*ft*****11H-WWH-DJHP0****(Di-h0-()ooooo0ftl-—Ql>J1’JMHHHH*U]**W*M*HHll-oMJ’JJMhhHQCD.HH-<CD(DCDU]WU]iWM)E’J1’JMlJCDllO**********H-fl<ftu1gU1*CDQQQ*******OH-00000000000CDOH-U1COHWl))CflH-CD•0ftQ——-------—-CD.H-H<LJ-Cl)J--JoU]LDH-ftLUQQHCl)Cl)C/)Cl)Cl)Cl)Cl)iH-aWU](DMWU1CDCI)H-H-l))Lfl0---0DlfrQMMMDUHHHHftLL-OftftCDH*******U)*—M--J0DU]DU1Wl))U1ftCD—PlDlHl-iCDHHCl)Cl)Cl)Cl)Cl)Cl)Cl)W)a)LoJ—JH0dH-H-OM]CDCDCDU]i/LUUiLU0HCI)0ft1MiftMiQHDMiU)0—Q—DlH-U]H-QjMitiQ—---—h0000000—Cl)ilH-ftH0H-——0MiHH-<H-CDH-CD<*Cj)——H-—ft000(0CDCl)H0OM—frH-ftOH(DhiHOCD0Dit1CD0UI(DCDH-C)H<H-H-hiH-MiU)CDCDCDDlLlQiH-ftl-*---Q—<CD0LiH-H-H-Cl)*—HH-ftCD—0DlH-Ci)*Cl)0DlQH*—LJ.H-ft(DCl)0t5-—H-—CDIC)CDhiM—I-iH-0ftZi0CDhCD>fthi-3QiH-hi0HH--CD—0CDMiOCDCl)ft*—Q—ftMU)HhiCl)lI——CDOCD<C)H-LLflQjCDX-—0H-<hi<-—CDLi.——MH-hiMCDCDLI.H-0LIhiH-CDrt—.Q.——0CDOhiO-CD—H-H-CD.hi—H-0HH-—0H-ft—:ftCDhhO—.fthi0Cl)H-0<i-H-H-0CDCD—hi(0ftHH-hihiQiCD00DlDlCDH-ft0—CDO(D<*Mi—0CDCl)CDCDiLLH[-liD)H-—HCDhihiftOH-000CD——-Mi0 hi2Fd**cthF-3DJOH-c1cctrtF--ct(flt’i[-I-I-I-L-hçtrtH-H-H-H-H-c-rH xOCO0COCOCOHCI2HIIOUW’)F-ij()IIIIIIIfCOf)COCOCOCObH—HHHi.<t-krtU]WMHO—II—1hCOCOCOCOCOCOL’iOCDHMOrOMt’JQ0<HOCD—I)COCOCOCOrrfJwwwwwc11WD.)H)COCD—c-cHQ.COCOCOCOCOCOHH0-.LfliW)Hoo—<COCOCOCOCOCOP)u1u1u1u1UiI--UiW)HH-—CDCOHCD CD H**I-F-3DJH-J(Ii(Dc-cHODJctH*tt(D(D(D(D(D(D(DIl-i-If-I—-4COCOCOCOUiUiWIIIICOCOowouiU1 wUiUi**COCOIW**COCOU]U]COCOCOIIIIICO7COwoJUiw * CO * CO-4


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