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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 COLUMBIA  by TRENTON JOHN HAGGARD B.A.,  The University of British Columbia,  1992  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE  in  THE FACULTY OF GRADUATE STUDIES (Department of Agricultural Economics)  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA  September 1994 -  Trenton John Haggard,  1994  presenting In this thesis partial in fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of department my or his by or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  (Sign  QjJ5 4 Department of ,ç  4D/ii?j  The University of British Columbia Vancouver, Canada Date_________  ii ABSTRACT  The production of forage in British Columbia plays and integral role in sustaining livestock herds within the province.  Forage is an important component  in the daily feed requirements of horses, sheep, and cattle. availability of considerable conditions  forage  costs  can  due  on  farmers  significantly  certain regions,  who  limit  shortfalls  (WSD)  raise the  conditions  livestock.  availability  in  in British Columbia,  production;  increases in the price of forage. Drought  or bad weather  can  Wide—spread of  forage  impose drought  crops  within  causing prices within those regions to become inflated.  Under standard insurance against  drought  to  Fluctuations in the  there  is  no  farmers  are only insured  compensation provided against  For those purchasing forage,  a Wide—Spread  insurance scheme would provide insurance against the price—risk  associated with drastic weather conditions.  However,  since forage prices are  required to operate such a policy and are non—observable, a mechanism is needed in order to estimate them.  A regional spatial price—equilibrium model which  relates regional prices to regional production is developed in this thesis.  The  model will eventually be used to predict prices and hence determine whether a particular region is eligible for a payout under the WSD insurance scheme.  A key  assumption behind the model is that according to the ‘Law of One Price’; prices are perfectly arbitraged. individual welfare,  In a competitive setting,  in which agents maximize  total welfare is maximized and prices between regions will  not differ by more than the transportation costs. This spatial price—equilibrium model is applied to British Columbia forage production.  The  Central Interior,  regions  incorporated in  Cariboo—Chilcotin,  the  study include the Peace River,  Thompson—Okanagan,  and Kootenay Regions.  The Lower Mainland/Fraser Valley and Vancouver Island are excluded as they do not typically fall under the forage crop insurance plan in British Columbia.  iii Table of Contents page  Abstract  ii  Table of Contents  iii  List of Figures  iv  List of Tables  v  Aknowledgements  vi  1.  1  Introduction 1.1 1.2 1.3 1.4  Background Problem Statement Study Objectives Organization of the Study  1 3 5 6  2. Review of Literature  7  2.0 Summary  7  3. Methodology and Model 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7  Overview Methodology Demand Curve Kinked Demand Curve Spatial Aspects Storage Mathematical Model Simulation Process  4. British Columbia Wide—Spread Drought Insurance Scheme for Forage Production : A Case Study 4.0 Introduction 4.1 Application 4.2 Computer Algorithm 5. Results 6. Conclusions 7. Bibliography  12 13 17 20 20 21 23 31  33 33 33 37 40 56  .  .  58  8. Appendix 1  60  9. Appendix 2  65  10. Appendix 3  67  iv List of Figures  page 1.1  Wide—Spread Drought compensation payment  4  3.1  Description of a region  13  3.2  Welfare maximization and the ‘Law of One Price’  16  3.3  Regional demand curve for forage  18  3.4  Demand curve for storage  23  3.5  Benefit measure under upper slope of demand curve  24  3.6  Benefit measure for points under lower slope of demand curve  25  4.1  Regional breakdown for British Columbia  34  5.1  Price/quantity plot for Peace River Region (Base Case Scenario)  42  5.2  Price/quantity plot for Central Interior Region  5.3  Price/quantity plot for Cariboo—Chilcotin Region  (Base)  43  5.4  Price/quantity plot for Thompson—Okanagan Region  (Base)  44  5.5  Price/quantity plot for Kootenay Region  5.6  Predicted over real price for Peace River Region  50  5.7  Predicted over real price for Central Interior Region  50  5.8  Predicted over real price for Cariboo—Chilcotin Region  51  5.9  Predicted over real price for Thompson—Okanagan Region  51  (Base)  (Base)  43  44  5.10 Predicted over real price for Kootenay Region  52  5.11 Price/quantity plot for Peace River Region  55  (Base Case)  v List of Tables  page 1  Regional livestock numbers  61  2  Regional base herd feed requirement  61  3  Yield per acre for forage crops  62  4  Seeded acreage and average regional production  62  5  Coefficient of variation/correlation coefficient matrix for regional precipitation  63  6  Cholesky’s decomposed matrix for production  63  7  Regional transportation cost matrix  64  8  Regional storage costs  64  vi Acknowledgements  I would like to thank the members of my committee: Dr.  Casey van Kooten,  and Dr. W. Waters.  Dr.  A special thanks to Jim,  advisor, for all of his help and effort towards this thesis. the guys at the Ministry of Agriculture, Branch) :  James Vercammen,  Fisheries,  my primary  Thanks also to the  and Foods  (Crop Insurance  Richard Scott, Lonie Stewart, Wayne Lohr, and Bob France.  Without the  funding and support from the Ministry and its members, this study would not have been possible. Thanks  to  all  my  Agricultural Economics, to the thesis.  fellow  students  and  members  of  the  Department  of  for putting up with me and providing occasional insight  A special thanks to Jodie for the use of her broad vocabulary and  superior editing skills. Mostly, I would like to thank my family for their never—ending support and encouragement.  Yeh, this thing is finally done...  1 Chapter 1  :  Introduction  1.1 Background Forage is an important crop in British Columbia as it contributes to the British Columbia livestock industry.  Livestock producers use forage as one of  the main ingredients in the composition of livestock feed.  Forage in some cases  comprises 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 intake  for cattle differs depending on type and sex of animal; however, a rough estimate of forage consumption by cattle would be approximately 30 lbs per day, sheep,  6 lbs for  and 16—20 lbs for horses.’ Relative  the  to  land  devoted  to  significant crop in British Columbia. 2 its growth in 1991, barley,  and canola.  its  production,  forage  is  a  highly  More than 800,000 acres was devoted to  compared to slightly over 100,000 acres for each of wheat, The largest allocation of land devoted to forage occurs in  the Thompson—Okanagan Region under which approximately 50 percent of the land is irrigated.  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 (Statistics Canada #95—3935,  1991)  On average, needs.  the province tends to produce enough forage to meet its own  In a typical year in which average yields occur, both the Peace River and  Kootenay Region are relatively self—sufficient requirements)  .  (producing enough to meet their  The Cariboo—Chilcotin Region produces less than it requires and,  as a result, crops will flow in from the Central Interior and Thompson—Okanagan regions  .  1  For further information refer to Keay (1991), National Research Council (1989), Agriculture Canada (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.  a In British Columbia, the majority of forage crops are produced by those who utilize it  (for livestock feed)  These farmers commonly store some of their  .  crops for use in following seasons.  However,  in times of drought  (periods of  unusually low levels of precipitation), this is often not enough to meet their herd’s forage requirements. or from other regions  As a result,  farmers will purchase forage locally  (other parts of the province, Alberta,  a price which is based on regional supplies and demand.  or Washington)  at  Given that forage has  no close substitutes in feed use and is relatively expensive to transport between regions,  forage demand tends  to be quite inelastic.  This  inelastic demand,  combined with highly variable yields and quality (due to variability in moisture and natural inputs in production),  results in a price of forage that is rather  volatile. Since the majority of livestock producers in British Columbia are both producers and consumers of the crop, prices  fall,  shortfalls  in  but  when  yield,  they and  rise  as  a  they are not only concerned when forage  as  Higher  well.  result,  farmers  prices  must  face  often both  relate a  to  reduced  availability of forage and an inflated price in making up the shortfall.  A fall  in the price of forage is often associated with an excess supply, and given that farmers typically consume the crops they produce, the situation is not as severe. Forage prices vary inversely with yield levels can  occur  as  frequent  as  one  in  five  6  and because wide—spread droughts  , 7 years  price—risk  is  an  important  consideration.  is commonly transported via truck, as it is the most available and convenient method of transport. It is not uncommon for forage prices to rise 50 percent above the average during a wide spread drought. 6  For example, a 20 percent decrease in yields below the average would result in a 20 percent increase 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 Statement standard  Under  crop  insurance  insured/compensated for shortfalls level  (some percentage of  average price level.  in  British  Columbia,  in production that  average production),  farmers  are  only  fall below a guarantee  with the  losses valued at  an  This means that compensation or indemnity payments equal  the shortfall in production below a guarantee level multiplied by the average price  level.  Farmers  who  must  purchase  forage  are  not  well  covered  shortfall year because forage prices tend to rise in shortfall years. been proposed by the British Columbia Ministry of Agriculture,  in  a  It has  Fisheries,  and  Foods that a WSD insurance scheme be designed to address this price—risk facing farmers. Under a WSD insurance scheme, farmers would be insured against the rise in price of forage due to wide—spread drought, with indemnity payments equalling the shortfall in production below a guarantee level multiplied by the difference between the market price of forage and a price trigger if the former exceeds the 8 latter.  A wide—spread drought would be necessary but not sufficient to trigger  a payment from this scheme. in nearby regions,  This is because when adequate stocks are available  stocks would flow in to alleviate the shortage and the price  in the shortfall region would not rise above the trigger. To  qualify  for  standard insurance level)  a  WSD  (i.e.,  payment,  farmers  would  have  to  eligible  be  if their actual production falls below the guarantee  and have regional forage prices exceeding a threshold price level  exceeding however,  some a  percentage  problem  with  of  the  insured  implementing  measurement of the actual exists for forage crops,  for  forage price  a is  value/average policy  such  required.  prices are non—observable.  as  price) this  Since no  .  one,  (i.e.,  There  is,  as  some  formal market  This means that without  some mechanism for determining the actual price levels, the values of indemnities  8  The price trigger level would be some percentage above the average price.  Transactions regarding the sale of forage occur privately between farmers, and prices vary depending 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. constructed, a  this model can then be used for the purposes of crop insurance,  mechanism will  current  Once  levels  available  be  of  forage  for  estimating  supplies would be  forage  prices.  Specifically,  incorporated with the  results from the study to generate regional forage price estimates. price estimates and the observed production levels,  as  regression Given these  the insurers can determine  the level of indemnity payments under the Wide—Spread Drought insurance scheme. Figure 1.1 represents a region for which an indemnity payment will occur. actual below  (estimated) the  The  price exceeds the price trigger level and production falls  guarantee  level.  The  level  of  indemnity  is  shown by the  shaded  rectangle.  Price Level  Et; mate4  Price exceeds -  Act&l  P(a.Lt  Price Trigger  Shortfall in Production  Guarantee Level  Quantity Level Figure 1.1  Wide-Spread Drought compensation payment.  5 1.3 Study Objectives The main purpose of this study is to devise a theoretically acceptable and potentially useful method of estimating the price of forage.  Given the problem  of trying to establish prices in different areas for a good that flows within and between these areas, incorporating spatial dimensions into the model is required. A regional rather than individual agent model is used because the WSD insurance scheme will be based on regional production and not farm—level production. Since the market for forage is assumed to be competitive and as a result prices in the regions will not exceed the transportation costs between them, the model utilizes the rational,  ‘Law of One Price’  assumption.  The fact that farmers are  profit maximizers and crops are able to flow freely between regions  ensures that the  ‘Law of One Price’  will hold.  The characteristics of each  region will be based on representative agents within that region. the  ability of each region to place production in  storage  Furthermore,  for use  in future  periods will also be incorporated. There are two main components to the model, the first one being that given a  set  of  observations  on  regional  supplies  of  forage,  it  will  show  the  equilibrium allocation for those quantities and the set of equilibrium prices. The second component allows for simulations to be run; production  levels  can be made.  The  that is,  random draws explicitly account  different production variances and covariances across regions. components, prices  randomly drawn for the  Combining both  production levels are drawn and equilibrium allocation levels and  solved for.  This can be done many timesin order to get a series of  equilibrium prices associated with the simulated quantities.  Regression analysis  is used to draw relationships between regional prices and quantities.  These  results can then be used in an insurance scheme, where, given regional quantities of forage, prices can be forecasted.  10  10  Although econometrics are used in this study, the model presented is a simulation model and not an econometric one. The econometrics are done on simulated and not real data.  6 1.4 Organization of the Study Chapter 2 provides a review of the literature and is followed in Chapter 3 by the methodology used in this study. of the model and the assumptions held.  Chapter 3 continues with a description Chapter 4 presents an application of the  model to British Columbia forage production and includes some description of the data used and its sources.  Chapter 5 and 6 follow up with a description of the  results, conclusions, and recommendations regarding the model’s application. Appendix  contains  analysis results,  a  description of the data used and generated,  and copies of the computer algorithms used.  The  sensitivity  Chapter 2  : Review of Literature  2.0 Summary Assuming 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 to  identify past literature that had approached the problem of estimating forage prices  (or of similar crops),  and which had simulated a spatial allocation type  setting. Prior to 1984,  no published studies relating to the estimation of forage  prices could be found.  A study by Blake and Clevenger (1984) noted the same, and  found only one unpublished study by Myer and Yanagida 11 topic.  (1981)  relating to this  Blake and Clevenger stated that the Myer and Yanagida’s paper combined  an estimated demand function for alfalfa in 11 western states with a quarterly ARIMA model to forecast quarterly alfalfa hay prices. paper,  however,  The Blake and Clevenger  developed a slightly different model that  alfalfa hay prices before the first harvest,  for the state of New Mexico.  used a two step procedure that linked an annual model, which seasonal price patterns seasonal price patterns.  start,  to  forecasted monthly They  forecasting the point at  a monthly model  that  identified the  They incorporated the estimation of a series of monthly  autoregressive price forecasting equations,  an annual alfalfa demand equation,  and an annual autoregressive acreage forecasting equation.  These results were  then used to predict monthly alfalfa prices for the state. In 1987, Blank and Ayer created an econometric model of the alfalfa market for the state of Arizona.  A similar study by Konyar and Knapp  an analysis for the aggregate California market. Knapp  (1990)  (1988) provides  A later study by Konyar and  incorporating much of their previous research, presents a dynamic  spatial price—equilibrium model of the California alfalfa market. was  used  to  forecast  alfalfa  acreage,  prices  paid  Their model  received,  and  transportation flows for the short and long run under base year conditions.  The  “This Myer and Yanagida study was later published in 1984.  and  8 base year results were then used for comparison in determining the effects of reductions in federal water subsidies and the implementation of a cotton acreage— reduction program. There are many other studies, aside from those focusing on price estimation of agricultural crops, which have focused on spatial allocation and pricing under the spatial allocation setting.  One common assumption made in many of these  studies is that in a competitive,  spatial environment in which goods can move  freely from one agent to the next, the ‘Law of One Price’ holds.  There are some  studies that may lead one to question the appropriateness of the ‘Law of One Price’ hypothesis, such as Ardeni  (1989) which showed that some of the evidence  to support the existence of perfectly arbitraged commodity process in the long run,  is  flawed due to inferior use of econometric techniques.  Other studies  counter these attacks, like Baffes (1991) who states that the ‘Law of One Price’ still holds  and any contrary evidence relates accounting for the transaction  costs as the failure.  Regardless,  the  ‘Law of One Price’  hypothesis will be  maintained within the current study. A competitive spatial equilibrium setting is simulated in the study by Liew and Shim  (1978)  They take the theoretical problem of maximizing an arbitrary  .  net welfare function.  It is reduced to the Dantzig—Cottle fundamental problem,  which is less complicated than the simplex tableau method as additional variables outside  of  solutions.  the  original  problem  are  not  necessary  for  They discuss the economic implications of dual,  obtaining  feasible  slack and surplus  vectors and the welfare maximizing marginal transformation of demand and supply among regions.  A similar concept to ‘The Law of One Price’ is assumed, in which  the price of the kth commodity in region  j  should not exceed the sum of the  transportation costs required to deliver that commodity from region i to the supply price of the kth commodity in region i.  j  and  A numerical example of the  model is also provided. A paper by Willett (1983) incorporates a typical competitive spatial price— equilibrium represented.  model,  with  both  a  one  commodity  and  multi—commodity  This is all done within a linear programming framework.  setting  The study  9 further examines and tests the theoretical conditions on prices and quantities, within  ‘Duality Theory’,  for a competitive spatial equilibrium solution to be  obtained. A  study by Beckmann  (1985)  offers  an  interesting  spatial pricing under two separate pricing techniques.  look  at  competitive  The effects of changing  transportation costs, size of fixed costs, and consumer density on the radius of markets,  under both techniques is examined.  profits and welfare were examined. within  provided  this  study  does  Further,  the effects on agents’  Unfortunately, the majority of information not  relate  directly  to  the  establishing a competitive spatial price—equilibrium type model.  problem  of  Only specific  effects that changes in parameters have on the overall solutions are identified. In Takayama and Labys’ a  spatial  allocation  (1986) study, a general overview of analysis within  environment  is  analysis  international  spatial  equilibrium  commodity  shown,  followed  is  by  presented.  the  An  between  general  interregional spatial equilibrium model.  example  two  of  a  countries  description  of  typical with  a  one  typical  A comparison between the use of the  quadratic programming method and linear complementarity programming method in a static spatial equilibrium framework is then made. made  to  some  of  minerals use.  the  recent  models  constructed  12  for  Furthermore, reference is agriculture,  energy,  and  These basically describe the new techniques used by some of the  main agents within these sectors. Another general overview of spatial economic theory is presented in the book  by  ‘Spatial  Harris  and  Theory’,  ‘Location Theory’  Nadji  then .  (1987)  refers  .  to  It  begins  spatial  with  a  general  equilibrium models  a non—equilibrium dynamic theory is described.  Finally,  transition  applied  the  of  in  to  relation  It explains that many of the spatial equilibrium models are  partial equilibrium special cases of the general theory.  of  description  theoretical  framework  to  an  The general system as a discussion of the model  is  presented,  including a description of the construction of and equations associated with an  12  A dynamic type framework was not presented in this study.  10 applied location theory model. Although there have been few studies done on the pricing of forage  (or  similar products), much literature exists on the creation of competitive spatial price—equilibrium models. are  directly  applicable  The techniques used in the majority of these studies to the  beginning that a regional, be  and  used,  the  current  paper.  competitive,  studies  shown,  It  was  established  from the  spatial price—equilibrium model was to  provided  a  general  basis  for  model  the  represented in this study. The current study’s model typically assumes regions to be both producers and consumers of forage,  and allows for forage crops to flow freely within and  between regions depending on transport costs, availability, and regional forage requirements.  It is a result of this competitive setting that the ‘Law of One  Price’ assumption can be made. price—equilibrium models,  The current model is similar to the other spatial  especially the one used in Konyar and Knapp  (1990)  They assume a competitive market exists for the good and that it can flow freely between 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, a direct link between individual periods. an  acreage  response  characteristic,  in which there is  The Konyar and Knapp model contains  where  major  producing  regions  have  fluctuating acreage depending on acreage from previous periods, expected prices received,  and yields.  The current study does not make any reference to acreage  response, as individual agents are assumed to be price takers and produce forage, independent of expected prices. no  acreage  Carry—over  response stocks  Making no direct link between periods and having  characteristic,  are  included  in  the the  current model study but  each  is not period  truly dynamic. is  treated  as  independent of the others. The Konyar and Knapp study, like the majority of spatial price—equilibrium studies  assumes  simplicity.  linear  This  may  inverse be  a  demand model  curves  for the  mispecification  good in when  question,  modelling  for  forage  11 production. there  are  Since farmers have specific base feed requirements to meet, costs  to  adjusting herd sizes,  the  individual  farmers’  and  demand for  forage will not fluctuate given small price variability around an average price level.  Therefore, an inelastic portion to the demand curve is needed around the  average price level when modelling individual farmers.  will not necessarily have the immediate upper and lower kink in the  however, demand  A regional demand curve,  and may  be  slightly  13  smoothed.  curves were incorporated into the  Nonetheless,  current study,  kinked  regional  demand  capturing the reluctance of  farmers to alter their base herd size. The  current  study,  unlike  econometric model.  Production  spatial  of  allocation  Econometrics mechanism.  is  data  supplies  then used on  others,  the  is  and  these  uses  randomly their  results  a  simulation  generated,  associated in  order to  and  not  an  with the optimal  prices  calculated.  formulate  a pricing  In the Konyar and Knapp study, actual production data are used.  They  also use econometrics in creating the pricing model, however, their results are not simulation—generated as in the current study. model  Further, the Konyar and Knapp  does not exploit the covariability between regional production  (due to  common weather patterns) The  aforementioned  characteristics  of  the  current  study  provide  some  potentially useful techniques which can be added to the past body of research devoted  to  pricing  forage.  Studies  relating  to  this  topic  are  few  and  information regarding the prices of forage can benefit those involved in the production of it and those involved in providing crop insurance and other types of government assistance.  13  When the representation of forage production is more diverse within a region, the regional demand curve may contain a smoother upper and lower kink.  12 Chapter 3 : Methodology and Model  3.0 Overview This  chapter introduces a pricing model  broken down into two main parts:  for forage.  The model  can be  one part generates random regional production  and finds the optimal allocation of that production, and the second part performs this  over  numerous  associated prices  simulations  can be  so  created.  that  a  series  of  Regression analysis  relationships between the quantities and the prices.  quantities  and  their  is then used to draw  A more detailed description  of these parts is described below. The first portion of the model allows for regional production levels to be simulated.  This is done by randomly drawing regional production levels  normal distributions around their means.  14  Carry—over stocks are added to the  randomly drawn production levels to create regional supplies. levels,  from  Given these supply  the model optimally allocates quantities for an equilibrium solution.  This solution is reached by assuming that each region maximizes its welfare given its own demand and supply for forage, prices in other regions, and transportation costs.  As a result,  crops will flow within and between regions in order that  total welfare be maximized.  15  The second part of the model randomly draws production levels and solves for equilibrium solutions over a number of simulations.  The random quantity  levels  and their associated prices  (as determined by points on the  demand  curves)  each  quantities parameterize  and the  are  collected  prices  are  from  simulation  16 generated.  relationship between  such  Econometrics  regional  14  quantities  that  a  regional  series  of  used  to  is  then  of  forage  and the  Normal distributions are used in the random draws, since the associated data requirements are small and multivariate random normal draws are more easily obtained than those with other distributions. 15  Total welfare equals the sum of each region’s welfare.  16  Prices are also considered to be normally distributed.  13 associated regional prices  3.1 Methodology The  model  regional level.  analyzes  transportation  flows  and  price  fluctuations  at  a  Each region is described as being both a producer and consumer  of forage, and is characterized by a representative agent in that region.  It is  assumed that the representative farmer produces forage in order to feed his/her own base livestock herd.  The regional livestock feed requirements depend on the  number of livestock present and the feed requirements per animal. From figure 3.1 shown below,  a typical region has both a production and  consumption sector and, depending on current supplies and demand has a number of options in order to meet  feed requirements and maximize welfare.  If surplus  crops are present, the region can allocate stocks to storage for future use or ship to other regions.  In times of excess demand,  stocks can be drawn from  storage (if they are present) or shipped in from other regions.  The arrows show  the direction in which forage crops will flow.  [ Typic& Region 1 nui  rather Region Figure 3.1  Regi  Storage  Description of  region  14 Farmers are assumed to be rational profit maximizers. base herd size,  They have a specific  and will produce forage in order to meet the feed requirements.  When a shortfall in production occurs such that the base requirement cannot be met,  the farmer will consider purchasing forage to make up the shortfall.  reverse happens when there are surplus stocks.  The  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 sufficiently low level will the farmer move away from his/her base herd requirements. Farm level behaviour must be assumed,  since this is a regional model and  does not explicitly observe farm level actions. profit maximizers,  facing parametric prices.  Farmers are considered to be  They buy and sell forage in order  to maximize their individual profits and as a result, established where total welfare is maximized. individual profits individual welfares, maximization  allows  (welfare),  Since all farmers are maximizing  and total welfare is defined as the sum of all  total welfare is maximized. for the creation  model’s key assumption,  a competitive market is  of  This  assumption of welfare  an equilibrium setting in which the  the ‘Law of One Price’, may hold.  The ‘Law of One Price’  states that the price in any one region will never  exceed the price in another region by more than the transportation costs.  It is  the assumption of a competitive market that validates the ‘Law of One Price’ in this model. welfare,  In a competitive world, individual agents seek to maximize their own  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 of  another region by more than the costs of transportation), individual agents (and as a result the collective of agents) result,  will arbitrage on this high price.  As a  the price will fall until arbitrage is no longer feasible and the ‘Law  of One Price’ holds. welfare the  Therefore, it can be stated that when individuals maximize  ‘Law of One Price’  total welfare is being maximized.  holds,  and when the  ‘Law of One Price’  holds,  If the ‘Law of One Price’ is not holding, then  individuals are not profit maximizing and total welfare is not being maximized. The following diagram, Figure 3.2, presents examples which help to validate  15 that the  ‘Law of One Price’  will hold when welfare is maximized.  example occurs at point A for both of the regions. place,  with Region  1  producing  first  In this case, no trade takes  and consuming at  producing and consuming at point CA2.  The  the point  CAl  and Region  Assume that the ‘Law of One Price’  2 is  violated, 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 being  represented, total welfare can be determined by strictly looking at consumption levels under the value of marginal product curves consumes at CAl, curve,  areas 1,  shown by areas 7,  (demand curves)  Region 1  .  therefore its welfare can be measured by the area under its and 3.  2, 8,  9,  Region 2 consumes at CA2,  10,  and 11.  therefore its welfare is  The total welfare is measured as the sum of  these two welfares. When the regions are able to trade, crops will flow from Region 2 to Region 1,  since the agents can be made better off by this.  Trade will take place until  the point at which the ‘Law of One Price’ is no longer violated (the point where the  price  in  region  transportation costs)  PB1,  1, .  exactly  exceeds  the  price  in  2,  PB2,  by  the  It is at this point that trade can no longer make both  regions better off, since the price for which the crop is sold is equivalent to its marginal value in consumption. maximized,  since  any  other  ‘  It is at this point that total welfare is  allocation  of  crops  other  than  this  equilibrium  allocation will cause total welfare to decrease. The total welfare associated with the second case has to include both the value  from  sale  and  from  consumption,  since  trade  has  occurred.  Region  1  purchases 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 the  quantity  cost  shown  difference  between  CAl  and  CB1.  This  However, Region 1 now benefits from areas 4 and 5,  is  a  by  area  5.  and therefore, gains area 4.  The sale of forage from Region 2 to Region 1 means that 2 now consumes at the lower level of CB2,  17  and therefore,  loses the consumption welfare shown by areas  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).  16 10 and 11.  However,  (the price sold at)  the sale of forage benefits Region 2 by the value of PB2 multiplied by the difference between CA2  and CB2.  As  a  result, Region 2 gains a welfare amount equal to the shaded area above the demand curve.  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 the  regions will be such that total welfare will be maximized, Price’ to  will hold.  the  point  that  and the ‘Law of One  In this example, both regions are made better of by trading prices  mc  longer  differ by  the  costs  of  transportation.  Compared with the first example, Region 1 shows an increase in welfare equivalent to area 4,  and Region 2 gains an amount equivalent to area 10.  Region 1  Region 2 Price  Price  Demand Curve  PAl  Demand Curve  PB1  P82  PA2  OA1  CB1  Consumption  CB2  c12  Consumption  Figure 3.2 Welfare maximization and the Law of One Price’.  The previous example relates directly to the situation in which a drought in one region creates a shortage of crop and causes the price in that region to  17 increase.  Due to arbitrage from welfare maximizing agents, surplus regions will  ship crops into the drought region and reduce the level by which that region’s price will increase. drought region,  the surplus regions’ prices will rise and the drought region’s  price will fall. which prices  Since crops are flowing out of the surplus regions into the  in  As explained earlier, all  regions will not  flows of crops will occur to point at  differ by more than the transportation  costs.  3.2 Demand Curve Each region has its own demand curve for forage which represents the value placed on forage in that region given regional prices.  This value is represented  by the areas under the curves (the cumulative value of each unit of forage)  .  For  given quantities of forage, regional prices can be established by the respective points on the curve. The demand curve is a value of marginal product curve,  and points on this  curve refer to points where the price of the marginal unit of forage equals its value of marginal product in production of livestock. maximize welfare,  Therefore,  they will purchase forage up until the point where the price  of the last unit of forage equals the value of its marginal product. do not purchase forage up until this point, point below the demand curve, its value of marginal product. their  own  when agents  welfare.  If  all  If agents  welfare will be measured at some  where the price of a marginal unit is lower than In this case the agents would not be maximizing agents  maximize  welfare  then  total  welfare  is  maximized and regional prices can be determined by the respective points on the regional demand curves. The shapes of regular demand curves are typically assumed to be continuous and downward sloping. shape of curve to use.  However, when modelling forage this is not an appropriate Since forage crops have a lack of close substitutes, and  farmers do not readily alter base herd sizes given small fluctuations in forage prices, sloping.  the shape of demand curve is not necessarily continuous and downward  18 Referring to perfectly inelastic  figure  3.3,  (vertical)  the  shape of  a regional demand curve will be  for a given range of reasonable forage prices.  Since farmers have specific stock requirements which they must meet, are  costs to  adjusting herd size  (i.e.,  and there  actual physical costs of buying and  selling cattle and the uncertainty associated with it),  the demand for forage  will not fluctuate given low variability in prices around an average price level. The  height  of  this vertical portion  of  the demand curve will  distribution and scale of the different farms in the region. farmers are operating at the margin, at the top.  depend on  the  If the majority of  then the vertical portion may kink sooner  These types of farmers will be more responsive to increases in hay  prices since their scale of operation is not as profitable  (flexible)  at the  margin  region  widely  as  other  distributed,  farms.  If  the  scale  of  farms  in  the  is  then the upper kink may become smooth as the farms at the margin  respond to small price fluctuations and the other farms gradually respond to larger price fluctuations.  Price of Forage  Upper Kink Point  Inelastic Portion of Curve  Lower Kink  Base Herd Feed  Forage Consumption  Requirement Figure 3.3  Regional Demand Curve for Forage  19 The regional quantity demanded for forage will likely respond if forage prices become excessively high.  Under a wide—spread drought scenario, with below  average precipitation across a large area causing decreased forage yields, there may not be adequate stocks in nearby regions to meet the excess demand for forage in that region.  As a result, forage prices may become so high that farmers begin  to decrease their consumption of forage.  This would occur when it is no longer  feasible for farmers to maintain and feed their present base herd size given the higher  forage prices.  The result is  farmers seek alternatives grains like barley,  a decrease  in the demand for  forage as  (i.e., reduce their base herd size; use more of other  if possible;  and feed less hay per animal)  .  sell off feeder calves earlier than expected;  If the drought is more severe and prices rise to  a higher level, the demand for forage will be even less and the demand curve will continue to slope backward to the left. The demand curve may be kinked at the bottom of the vertical portion since exceedingly low forage prices may entice farmers to increase their base herd size and forage usage.  Given farmers present base herd size and feed requirement (as  represented by the vertical portion of the demand curve) when forage prices fall low enough  (eg.,  favourable weather patterns across large areas causing surplus  forage yields and large drops in forage prices), farmers may choose to invest in livestock.  Given the drop in prices,  farmers can meet the feed requirements of  a base herd larger than the present herd. debated,  Although this bottom kink can be  it is the upper kink and slope that are of interest for the model  (the  price responses to drought situations on the upper portion of the demand curve) A graphical description of a regional demand curve would assume a linear curve intercepting the vertical axis and sloping downward to the right. base herd feed requirement demand curve becomes portion.  level  At the  (as described on the horizontal axis),  downwardly vertical.  This  represents  to the  this  inelastic  At the bottom of the inelastic segment, the demand curve slopes down  toward the right again.  20  3.3 Kinked Demand Curve When using a continuous downward sloping demand curve, it is not necessary to know the section of curve being utilized because one function represents all points on the curve.  However, non—continuous (kinked) demand curves require that  the section of curve be identified.  This is due to a separate linear function  used for both the upper and lower slopes of the curve.  If optimization occurs  to 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 of the curve is used.  The upper portion of the  separate parameters  than the  downward  In  intercept  sloping. (au,  slope value  (1)  form  it  The upper portion is  linear and  can  a  be  described  as  vertical  referring to the upper curve), plus a negative  multiplied by the level of consumption  ()  (c’,  mathematical  the subscript ‘u’  horizontal axis) intercept  lower curves.  regional demand curves contain  (C,  represented on the  The lower portion of the curve is described as the vertical  .  the subscript  ‘1’  referring to the lower curve),  multiplied by the level of consumption  (C)  .  minus a slope  Note, that the method used in  identifying which section of the curve is being referred to is later described in a mathematical description of the model.  3.4 Spatial Aspects As  discussed  previously,  the  model  supplies within and between themselves,  allows  regions  allocate  to  and to and from storage,  forage  in order to  maximize welfare.  When dealing with continuous downward sloping demand curves  in  spatial  this  type  of  setting,  the  results  will  appear  (price/quantity relationships are based on continuous curves)  to .  be  uniform  However,  when  incorporating non—continuous (kinked) demand curves like those in this study, the results will differ  (price/quantity relationships are based on non—continuous  curves) When regions optimize welfare under a kinked demand curve,  the results  contain jumps when moving from one section to the other in the demand curve. vertical intercept  (and possibly the slope)  The  are different when referring to the  21 two separate portions of the curve to the left and to the right of the inelastic section.  This inconsistency in the demand curve will create different types of  solutions in which the ‘Law of One Price’ holds.  In the optimization process,  regions will not have the gradually decreasing marginal benefit as consumption increases.  Marginal benefit does gradually decrease to a point, as consumption  increases;  however,  an  increase  in  consumption  beyond  the  base  herd  feed  requirement results in a discontinuous fall in the marginal benefit (moving from the upper slope to the lower one) Regional prices may also exhibit some non—uniform results.  When regional  consumption levels are to the left of the inelastic portion, pricing is based on the upper slope.  And, when consumption is to the right of the inelastic portion,  prices are based on the lower slope.  But,  when regional consumption exactly  equals the regional base herd feed requirement, region’s  demand  curve,  inelastic portion. trade,  as  the  actual  price  pricing cannot be based on the level  is  somewhere  It is known from the ‘Law of One Price’  within  the  that when regions  flows will occur to the point that regional prices will no longer differ  by transportation costs.  Therefore, the region whose consumption equals its base  herd feed requirement can calculate its price from another trading partner’s price,  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’s price plus transportation costs, and when exporting forage, its price equals the other’s price minus transportation costs.  3.5 Storage Some  dynamic  characteristics  are  incorporated  into  the  model.  These  characteristics show that farmers at most will look one period ahead into the future, by allocating stocks to storage for use in that period.  Given farmers’  profit maximizing behaviour, they will arbitrage on forage prices through their use of storage.  Assuming that farmers are rational, they will determine whether  or not to store production for use in the following period given current forage prices, costs of storing across one period, and the expectation that next period  22 will witness an average price given average production. Not only are farmers able to send forage stocks into storage, but they also receive stocks from storage.  Therefore, when farmers determine how to optimally  allocate current production, they include the amount of stocks they are receiving from storage during that period.  The model forms no continuous link across time  between the amount of forage shipped to storage in one period and the amount of stocks in storage added to production in the following period, since each period is  treated  as  an  independent  simulation.  Given  the  model  has  no  formal  dependency across time, it cannot be considered a truly dynamic model, however, the aforementioned aspects relating to storage do provide some mildly dynamic characteristics to the model. A steady—state characteristic is necessary when including storage in this model.  On the average,  the carry—over stocks included in a current period’s  supply must equal the stocks allocated for use in the following period. this characteristic,  the results will be biased as  equal their true average values.  Without  current supplies will not  Steady—state for storage is incorporated into  the model by running the set of simulations a number of times, until on average stocks in equal stocks out.’ 9 Figure 3.4, storage.  presents a graphical representation of the demand curve for  The expected price next period is represented on the vertical axis with  stocks allocated to curve  intercepts  storage for use next period on the horizontal axis.  the  vertical  axis  and  is  linearly  downward  The  sloping.  An  interpretation of this curve would be that if no stocks are stored for the period following,  an average price can be expected for next period  point touching the vertical axis) following period,  19  As more and more stocks are allocated for the  the expected price next period will fall further and further  below the average price.  18  .  (referring to the  This relates to the curve’s negative slope.  Note that current forage supplies include carry-over stocks from the previous period.  Carry-over stocks are not included in the first set of runs since they must be generated a priori. After the first set of runs are completed, stocks in are included in current supplies for allocation purposes.  23  Price in  fotlowing period  Next period’s price driven to zero  Quantity stored for use next period Figure 3.4  Demand Cutve for Storage  3.6 Mathematical Model The  model  allocations  of  in  this  forage  maximize  total welfare  regional  forage  study  crops across  supplies  determines  (for  the  regions.  the  purposes This  optimal of  inter/intra—regional  consumption)  in  allocation process  and demand and on the  assumption that  function is being maximized subject to various constraints.  20  order  to  is based on an  objective  This objective  function represents total welfare across all regions, where total benefits (i.e., the cumulative area under all regional demand curves) costs The  (i.e., the sum of all transportation costs  following  section  presents  a  description  are subtracted from total  associated with crop flows) of  the  model,  including  the  objective function and constraints. One of the main purposes of the model is to determine optimal allocations  20  parameters for the regional demand curves are fixed throughout the optimization, while regional production for each scenario is randomly drawn ex ante.  24 of crops within and across regions. function The  (representing total welfare)  first  includes  portion  of  mathematical  the  into  the  subject to various necessary constraints.  objective  formulas  optimization  function  describing  described in the previous section, curves  This is done by maximizing an objective  (representing  the  regional  demand  benefits)  curves.  As  in order to incorporate the regional demand  process,  it  is  necessary  portions of the demand curves are being referred. simple task,  total  to  identify  Unfortunately,  to  which  this is not a  as will be shown.  When referring to a point left of the inelastic slope, the area under the curve as  (welfare benefit)  represented  portions:  on the  is measured up to the actual level of consumption horizontal  axis.  This  area  can be  the triangular area above the actual price  area below that price.  For a graphical description,  calculated in  (a,_r3*C)  see Figure 3.5 below.  Actual Price Slope = f3  -  U  f3  *  U  Actual Consumption Base Herd Feed Requirement  C Figure 3.5  Consumption  Benefit measure under upper slope of demand curve  two  and the square  Price xu  (C),  25 The following formula represents that area  (benefit)  under the curve:  Q5* [ (a_(a_*C) ) *c+ (a_*C) *C. ]  (3.7.1)  This is equal to one—half the difference between the intercept price and actual price, multiplied by the consumption level, plus the actual price multiplied by the 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. area to  the  left  of the  inelastic portion plus  consumption and base herd feed requirement level.  the  It includes all of the area between  the  See Figure 3.6 for a graphical  representation.  Price  CL  Actual Price  13’ 1 cx  -  13  1 a -  Base Herd Feed Requirement  *CB  -c  Actual  CB Figure 3.6  C  actual  Consumption  Benefit measure for points under lower slope of demand curve  26 area  The formula,  left  of the inelastic portion is  with the actual consumption level  requirement level (CB) inelastic  portion  .  and  (C)  represented by the previous  replaced by the base herd feed  The area under the lower slope of the curve, between the the  actual  consumption  level,  is  represented  the  by  formula:  *C)*(C Q5 i _ 1 1 13i*C)]*(C_ *CB 1 CB)+(a _CB) *[( )_(  (3.7.2)  This is equal to one—half the difference between the lower kink price and the actual price, requirement, Therefore,  multiplied by the plus  the  level  price  actual  at  which consumption exceeds  multiplied by  that  quantity  the base  difference.  the total area left of the inelastic portion plus the area included  to the right is represented by the formula:  1 *CBQ + 2 * ((C+C (CCB) 5**CB B)/2)J *[aj_  (3.7.3)  Given the demand curve has two distinct sections and the portion right of the inelastic slope),  ((i.e., the portion left  it is necessary to identify in  which section actual consumption falls when welfare is being optimized.  Using  the two previously described formulas for measuring benefit under the upper and the lower slopes of the curve, identifier parameters can be used for identifying where actual consumption falls.  Attached to the previously defined formulas in  the objective function are two identifier parameters ‘TA’  and ‘TB’  .  ‘IA’  is  multiplied by the formula relating to area measurement for points left of the inelastic  section,  and  ‘TB’  is  multiplied by  the  formula  relating  to  area  measurement for points to the right of the inelastic section. The two identifier parameters are unique in that they only represent a value of one or zero, ‘TA’  equals  one,  then  with neither having the same value. ‘TB’  equals  zero,  and only  the  In other words,  formula  relating to  if a  consumption level less than the base herd feed requirement level is represented in the objective function.  If ‘TB’ equals one, then ‘TA’ equals zero, and only  27 the  formula  for a point to the right is  represented.  These  identifiers are  successfully able to identify whether consumption falls below or above the base herd level by a series of constraints.  Included in these constraints are that  the 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  associated identifier parameters ‘IA’ portion  of  represents  the  objective  the  total  function  benefits  under  and ‘IB’ and  across  are all  demand  the  curves)  and  the  are included under the benefits summed  over  regions.  represents this portion of the objective function,  all  The  regions.  following  This formula  in summation notation:  (3.7.4) (C+CB) /2))].  Note,  that the subscript ‘u’  portion,  ‘1’  refers to the demand curve left of the inelastic  refers to the curve right of that portion,  and ‘n’  refers to the  region. The second part of the objective function relates to the costs incurred through  the  regions. regions (also  allocation  These  costs  (transportation) are  equal  of  forage  the per unit  to  crops,  within  transportation  and between costs  among  (represented as a matrix) multiplied by the quantity levels transported  in  matrix  notation)  .  When  these  two  matrices  are  multiplied,  they  represent the transportation costs matrix shown in the objective function, where the costs are represented by all of the trace elements. The transportation costs matrix is calculated by multiplying  the square  matrix of per unit transportation costs  (A) by the transpose of a square matrix  of transportation quantity levels (T’)  The per unit transportation cost matrix  .  is represented by each region for both the rows and the columns.  This allows for  all possible transportation combinations to occur, inciuding allocations within  28 regions.  The transpose of the quantity transported matrix is used since its  combination with the per unit cost matrix yields a transportation cost matrix in which the elements correctly match up.  In other words,  the correct per unit  costs are attached to the their respective quantity levels, thereby allowing for the costs to be represented as the trace elements in the transportation costs matrix.  In matrix notation, this transportation costs matrix can be represented  by the following formula:  trace (A*T’),  (3.7.5)  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 objective  function can be stated in the following formula:  (3.7. 6)  W  Er[IA* (*C_O 5*  =  .  in (C+CBr) /2) )  I  —  ) +IB* (a *CBO 5* 2 *C  + (C—CB) 2 *CB  *  trace (A*T’  This function represents total welfare across all regions, where the parameter W refers to welfare. of all costs.  It is separated into the sum of all benefits minus the sum  In the optimization process,  this function  (total welfare)  is  maximized subject to a number of constraints, which will be described below. There are ten constraints in the optimization process. constraint  that  all  of  the  choice  parameters  These include a  (regional  consumption,  transportation levels, and the two identifier parameters) be non—negative.  This  is represented as follows:  (1)  C,  T,  IAN,  IB  where consumption (C), transportation levels are included for all regions.  (T), and the identifier parameters  29 The second and third constraints are necessary for the allocation process. These include the constraint that a region cannot consume more forage than it possesses and gets  shipped in from elsewhere.  The following represents this  constraint:  (2)  2 + 1 T + T .  C,  where the first  .  .+T,  subscript refers to the destination region for the crop being  transported and the second refers to the source of that crop. The third constraint relates to regions not being able to allocate more forage than they possess  (including crops shipped in from elsewhere)  .  This is  shown by following formula:  (3)  2 + 1 T + T ..  QP  where QP refers to a specific regions supply level  (production and carry—over  stocks) The fourth and fifth constraints relate to shipments to and from storage. These are shown by the formulas:  (4) (5)  0 and 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. allocation  process  is  included  separately  in  the  program,  as  This  described  previously. 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.  30 These constraints are represented as following:  (6)  IA* (C—CB)  0,  (7)  IB*(C_CB)  0.  and  Another constraint ensures that the identifiers sum to one. (8)  IA+IB  =  This is shown by:  1.  The last two constraints force the identifiers to assume values not greater than one.  They are represented by:  (9)  IA,  1 and  (10)  IB  1.  For further reference, a summary list of the parameters used in the program is shown below: W  total welfare  =  =  vertical intercept for upper slope of demand curve  =  vertical intercept for lower slope of demand curve  13 13  slope of upper portion of demand curve =  CB  =  slope of lower portion of demand curve regional base herd feed requirement  (horizontal value of inelastic  portion of demand curve) C  actual regional consumption of forage  =  QP  =  regional  supply  of  forage  (includes  production  and  carry—over  stocks) A  =  vector of transportation costs within and between regions  T  =  transportation quantity levels of forage  IA  =  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 0 if not)  31 lB  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 0 if not)  3.7 Simulation Pzocess As described previously, the model used in this study is a simulation model and not  an  econometric  one.  It  randomly generates  performs its numerous simulations.  data by which the model  A description of this simulation procedure  will be presented below. The model begins by generating random correlated production levels for the regions. regions,  To get these values,  the model generates production levels for the  assuming no correlation between regional production.  To do this,  a  vector of average production levels of all the regions is added to a vector of independent, multivariate normal draws  (with means of zero and variances of 1)  This allows random independent quantities to be generated, with the mean values for each region’s production taken into account. The correlations between the random production draws is accounted for by multiplying this generated vector of regional production levels by a Cholesky decomposed matrix. 21  The Cholesky decomposed matrix is a nonsingular triangular  matrix that has the property that when multiplied by a vector of independent random normal  draws  Cholesky matrix  is  create  will  created by  regional production.  a  vector  of  correlated  random  draws.  The  decomposing the variance/covariance matrix  of  In simplicity, the variance/ covariance matrix is created  from a data series of annual regional production levels.  From this matrix,  a  matrix of characteristic vectors and diagnol matrix of characteristic roots can be found.  The two later matrices are multiplied,  and together,  have identical  properties to the nonsingular triangular matrix described above. Given optimal  21  these  random draws  allocations  for  forage.  See Judge (1988), p. . 4 494 96  for regional production, This  is  done  by  the model  maximizing  the  finds  the  previous  32 objective function subject to the constraints.  Note that the ‘Law of One Price’  is assumed throughout this optimization procedure, ensures that this law holds.  since welfare maximization  The determined regional consumption levels are then  related to the regional demand curves to obtain the regional prices. This procedure is performed over numerous simulations in order to obtain a series of regional prices and their associated quantities. (OLS)  is  then used to draw relationships between  Regression analysis  the prices  and quantities.  These results can further be used for the purposes of estimating regional prices given actual quantities. This insurance  pricing scheme  model  for  has  British  application Columbia  for  forage  use  in  a  producers.  Wide—Spread As  noted  Drought in  the  introduction, British Columbia forage producers face price risk associated with wide—spread drought.  It is for this reason that a Wide—Spread Drought insurance  scheme has been proposed,  and subsequently a pricing mechanism needed.  The  following chapter will present an application of this model to the pricing of forage in British Columbia for the purposes of a Wide—Spread Drought insurance scheme.  33 Chapter 4  Model specification for British Columbia  4.0 Introduction The  following chapter presents  British Columbia forage. of the data used, simulations.  an  application  of the pricing model  to  Included are a breakdown of the regions, a description  and an explanation of the computer algorithms to perform the  The following two chapters present some of the results obtained and  the conclusions regarding the model’s application.  4.1 Application The regions:  province  of  British  Columbia  was  broken  down  into  five  separate  Peace River Region; Central Interior Region; Cariboo—Chilcotin Region;  Thompson—Okanagan Region; and Kootenay Region, with each region considered both a producer and consumer of forage. British  Columbia  Ministry  primary  funding  agents)  of and  This regional breakdown was determined by  Agriculture, based  Statistics Canada #95—393D (1991) boundaries for British Columbia.  .  on  the  Fisheries regional  and  Foods  boundaries  (the  study’s  defined  in  Figure 4.1 shows a rough approximation of the  Piovirice of BnLish Columbia 34  Regional breakdown for British Columbia  Figure 4.1  The Peace River Region includes the Peace River Regional District.  The  Central Interior Region includes both the Bulkley—Nechako Regional District and Fraser—Fort George Regional District. Cariboo Regional District. Lillooet  Regional  Colurnbia—Shuswap  The Thompson—Okanagan Region includes the Squamish—  District,  Similkameen Regional  District,  Regional  The Cariboo—Chilcotin Region includes the  Thompson—Nicola  22  (Subd. A)  District,  Okanagan—  Central and North Okanagan Regional District,  District,  and  Kootenay Region includes the Central and Kootenay Boundary  Regional  Kootenay  Boundary  (Subd.  B)  .  East Kootenay Regional District,  The and  22  The model did not include any regions outside of British Columbia. Some forage is transported between Alberta and the Peace River and Kootenay Region; however, that link was excluded since British Columbia is a net exporter of forage (droughts in Alberta not having as much impact on forage prices in B.C. than if B.C. was a net importer) and the inclusion of Alberta would considerably complicate the problem (more regions to include, data to collect, and variables for which to solve).  35 Although the model in this study does generate its own data for simulation purposes, there are some actual data requirements that must be met in order for the model to become operational. forage,  production of forage,  and the costs of storage.  These requirements refer to data on demand for  transportation costs within and between regions,  The data and its sources are listed below.  Information relating to demand for forage was needed for the study. assumed  that  regional  consumption  (demand)  regional base herd feed requirements.  of  forage  is  It was  determined  by  the  These feed requirements are based in turn  on both the regional livestock numbers  and feed requirements per animal  (for  which 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 of  the Appendix and were obtained from Statistics Canada #95—393D  (1991)  Feed  .  requirements for each category of animal are found in Table 2 and were obtained from:  Keay (1991); Agriculture Canada (1986); Ross (1989); and National Research  Council (1989)  This table also includes the regional base herd feed requirement  .  values used in the study. Further parameterize estimates  for  information the  regional  forage  on  demand  were  province, see Aumack et al.  demand  curves.  obtained (1994)  .  for  from  forage Average various  was  needed  regional forage  rough  calibrated.  estimates  and  order  long—term  experts  to  price  around  the  These same individuals provided insight into  the responses of average farmers to fluctuations in forage prices. these  in  information  that  the  model’s  It was from  parameters  were  23  Production data was also needed.  This related to information describing  average regional production and the variance/covariance of production.  Average  regional production was calculated from data characterising regional yields per  23  No series data on forage consumption was available to obtain graphical estimates of the demand curves.  36 acre and seeded acreage. from:  The yield data is shown in Table 3 and was obtained  Statistics Canada #22—201 Annual Statistics and Grain Trade of Canada  (1990),  Statistics  Canada  #22—002  Agriculture, Fisheries and Foods  (1991),  (1994)  British  Columbia  Ministry  of  Seeded acreage data is found in Table  .  4 and was acquired from Statistics Canada #95—393D (1991)  .  Also, included is the  average regional production values used in the study. There was no series data available on regional yields, direct means for calculating production variability.  and therefore no  As a result,  some other  means was needed for calculating the variance/covariance in regional yields. assumption regional  that  the  variance/covariance  yields  was  made.  This  in  assumption  regional was  precipitation  relatively  An  affected  realistic  since  fluctuations in precipitation are the main impetus behind variability in yields. Therefore, of  given series of monthly regional precipitation levels,  variation  and  correlation  calculated.  Assuming  precipitation  and  a  relationship  variation  established for the regions.  coefficients  in  for  of  , 24 yield  a  regional  0.3  between  coefficients  precipitation the  were  variation  variance/covariance  in  matrix  was  The variance/covariance matrix was then transformed  via the Cholesky matrix decomposition method into a nonsingular triangular matrix  p,  which  was  then  used  for  randomly  drawing  regional  production.  25  The  coefficient of variation/ correlation coefficient matrix for the rainfall data is found in Table 5 and the triangular decomposed matrix is in Table 6. Transportation costs were also a necessary data requirement.  A matrix of  transportation costs per ton of forage within and between regions are found in Table 7.  These figures were calculated using transport cost quotes from various  trucking companies throughout British Columbia, see McConughy at al.  (1994)  .  The  average  distances  (1987)  The regional centres were chosen by the Crop Insurance Branch at British  .  between  regional  centres  is  found  in Ministry  of  Tourism  No studies or data were available to make this exact relationship. The value of 0.3 was used 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).  37 Columbia Ministry of Agriculture, intense areas  in each region  forage.  The  Fort St. John (Peace River Region); Vanderhoof  Williams Lake  (Thompson—Okanagan Region);  and represent the most  for production and consumption of  regional centres are as follows: (Central Interior Region);  Fisheries and Foods,  (Cariboo—Chilcotin Region);  and Cranbrook  Kamloops  (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 data requirement. spoilage.  Storage costs relate to the value of stored forage lost due to  The costs included in the study are shown in Table 8 and represent a  percentage of the stored forage that is lost from spoilage, value of that forage.  multiplied by the  Information on the costs of storage were obtained from  British Columbia Ministry  (1994)  and Soder  (1976)  4.2 Computer Algorithm In this section, the computer algorithms in the study are explained. main portion of  the computer  package, see Appendix 3. which the  output  is  calculations were made using the  GAIVIS  The  computer  The program starts by defining three separate files in  sent.  Three  sets  are then  defined,  labelling the  five  regions, the number of iterations or simulations to complete, and a set used in a loop to solve for price. demand  curves,  base herd  The parameters of the model describing the regional feed requirement,  entered using a series of ‘parameter’ The parameter ‘IND’ drawing  production  and mean  regional production  commands.  and the Cholesky decomposed matrix 2 used in randomly  levels,  are  entered.  Two  separate  definitions help to generate the random production levels. assumes  that  the  are  random production  levels  for  the  set  of  The first definition  regions  are  equal  Cholesky matrix multiplied by a vector of non—correlated normal (with mean 0 and variance 1),  parameter  added to the mean production levels.  to  the  random draws The second  assumes 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.  38 The parameters Si through S5 are used in storing carry—over stocks to be included times  in  in  next  the  order  (steady—state  for to  150  27 simulations.  current  be  supply  imposed  These  levels  on  to  simulations  converge  carry—over  stocks)  to  are their  A  .  transportation costs within and between regions is then entered. function  and  constraints  described  definition of the model ‘Versioni’ storage  previously  are  entered,  and the solve command.  repeated six true  matrix  values of  the  The objective followed  by  a  The allocations to  (for use in the following period) are saved in the parameter Si and used  in the second set of 150 simulations.  This is repeated a number of times for the  convergence described previously. The end of the program contains a set of ‘if—then’ statements to solve for prices.  If the optimal consumption level for a region does not equal the base  herd requirement, then price is based on the demand curve. equal the base requirement curve),  (i.e.,  If consumption does  it is on the inelastic portion of the demand  then price is based on another trading region’s price plus  transportation costs, depending on the direction of flow.  or minus  The final set of ‘put’  statements send the regional supply levels and their associated prices to the output files. Given the generated price and quantity series,  regression analysis is used  to define the parameters for estimating forage prices from regional supplies of forage. by  Prices are estimated for each region under each of the 150 simulations  substituting the  regional  quantities  back  into  the  regression  equations.  These estimated prices can then be used in determining levels of payments under the proposed Wide—Spread Drought insurance scheme. Since the slopes of the upper portion of the demand curves are not known with certainty and play an integral role  in the cost of the program  28,  three  27  Carry-over stocks are not included in the first set of 150 simulations, but are included thereafter. The steepness of the slope of the upper portion relates changes in consumption to changes in price and thus affects the estimated price levels and the cost of the program (i.e., an increase  39 different slopes are used to see the effect on prices. a  steeper  upper  slope,  the  level  of  estimated  It is assumed that with  prices  will  increase,  thus  increasing the frequency and level of payments.  in the level of estimated prices will increase the frequency and level of payments under the insurance scheme).  40 Chapter 5  Results  The first set of results were obtained with all of the upper slopes of the demand 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 was unavailable, a common slope was used representing an average of all upper slopes with an elasticity of negative one at the upper kink point.  29  Note that the  Kootenay Region was given a steeper slope than the common one since its upper kink point  (at the base herd feed requirement  vertical intercept.  level)  was much closer to the  In other words, it was necessary to give the Kootenay Region  a greater slope, since the vertical intercepts to the upper slopes of the demand curves should not necessarily exceed transportation costs between regions. Before the main results are presented, a typical year (simulation run) will be described to show the flows that can occur in the allocation process. was done using the ‘Base Case Scenario’  parameters.  This  In this typical year,  randomly drawn production levels for the regions are as follows:  the  143,857 tons  for the Peace River Region; 262,373 for the Central Interior Region; 104,265 for the Cariboo—Chilcotin Region; 146,815  for  the  334,676  Kootenay Region.  for the Thompson—Okanagan Region;  Note  that  the  Central  Interior,  and,  Cariboo—  Chilcotin, and Kootenay regions’ production levels include quantities drawn from storage; these values equal 68,194, the  regional  base  sizes,  herd  the  216,675 tons for the Peace River, the Cariboo—Chilcotin,  249,131  Kootenay  a  regions.  As  15530, and 61,232 tons respectively. regional  base  herd  feed  requirements  are  162,368 for the Central Interior, 208,734 for  for the Thompson—Okanagan,  result,  Given  both  the  Peace  River  and 63,461  and  for the  Cariboo—Chilcotin  regions 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 upper kink points was valid other than a general acceptance for assuming this by the Crop Insurance Branch at British Columbia Ministry of Agriculture, Fisheries and Foods, and after consultation and debates with hay suppliers and consumers in B.C.  41 an equilibrium solution will occur when the Central Interior Region ships 72,818 tons of forage to the Peace River Region and 27,187 tons to the Cariboo—Chilcotin Region; the Thompson—Okanagan Region ships 77,282 tons to the Cariboo—Chilcotin Region; and, the Kootenay Region ships 83,353 to storage.  The result is that all  regions are able to exactly satisfy there base herd feed requirement levels, with the Thompson—Okanagan Region consuming 8,261 in excess of its base requirement level.  Note that the differences in regional price levels in this equilibrium  solution do not exceed the costs of transportation,  where the trading regions  prices exactly exceed transportation costs; the price in the regions equals 111 dollars  per  ton  (Peace  River  Region),  69  (Central  Chilcotin), 72 (Thompson—Okanagan), and 41 (Kootenay) Region exactly exceeds the storage price 62 cost of storage On  produces  of  forage  and  requires  significant net importer of forage, The  exporters, 162,368,  The price in the Kootenay  .  tons  Central  with  the  Interior  Central  The Peace River Region produces  216,675,  and  the  Kootenay  Region  The Cariboo—Chilcotin Region is a  producing only 111,893 tons but requiring  and  Thompson—Okanagan  Interior  producing  regions  193,839  tons  are and  net  both  consuming  and the Thompson—Okanagan producing 318,272 and consuming 249,131.  As a result of this mismatch in supply and demand for forage, flow between and within regions.  demand by  crops will  Note that storage is excluded from this example  as steady—state for storage is assumed. excess  by the  the Peace River and Kootenay regions produce approximately  77,925 but requires only 63,461.  208,734.  (Cariboo—  (due to spoilage)  average,  tons  94  (expected price next year)  enough forage to satisfy regional requirements. 195,333  Interior),  importing 21,342  tons  The Peace River Region satisfies its of  forage  from the Central  Interior.  Cariboo—Chilcotin Region imports 10, 129 from the Central Interior and 83, 605 from the 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 created.  the  ‘Base  Case  Scenario’  parameters,  150  The results can be seen in Figures 5.1 to 5.5.  simulation  runs  were  The graphs show a plot  42 of  each  region’s  predicted  30 prices  on  the  region’s  supply  included are the average predicted price and production levels. typical  simulation  for  the  ‘Base  Case  Scenario’  was  shown  in  levels.  Also  Note that a the  previous  example.  160 140 120  1100  80 60 40 20  0 0  50000  100000  Figure 5.1  °  150000  200000  250000  Regional Supply Peace River Region (Base Case Scenario)  The predicted price values are similar to the actual prices. A description of the method used for predicting prices will follow later.  300000  43  120 I.  100  Average Price  m 80  •  I  60  •  III..  ii•  • S  •  40  Average S  20  0 0  50000  100000  Figure 5.2  150000  200000  250000  300000  Regional Supply Central Interior Region (Base Case Scenario)  140 SI  Average Price  120  100 80  60 0  40 Average Production  20  0 0  50000 Figure 5.3  -  100000  150000  Regionai Supply Cariboo-Chilcotin Region (Base Case Scenario)  200000  350000  44  120  Average Price  100  80  1:  I a  a  I  I  .1  20 Average Production  0 0  50000  100000  150000  Figure 5.4  200000  250000  300000  350000  400000  450000  500000  Regional Supply Thompson-Okanagan Region (Base Case Scenario)  80 70  Average Price  Is II  I  60  ph  50  I  I II  a  Ia  a  II  Is  a  a  •  a  I  III  I  I  30 a-  II  I  a I  I  20  a  •  10 0 0  20000  40000  60000 Figure 5.5  80000  100000  120000  Regional Supply Kootenay Region (Base Case Scenario)  140000  160000  180000  45 From the graphs, predicted prices appear to rise as production levels fall as shown by the downward sloping trends. regional  production  levels,  and  Since predicted prices are based on all  only  one  region’s  production  levels  are  represented 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 impact of  other  regions’  production  levels  on  the  price  levels  of  the  region  in  question) The predicted prices for the Peace River Region ranged from a low of 35 dollars per ton to a high of 160, given an average of 108.  The supply levels for  the Peace River Region from the 150 simulations ranged from 110,000 to 290,000 tons.  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.  The  Cariboo—Chilcotin Region had prices ranging from 25 to 135, given an average of 96, with supply ranging from 70,000 to 150,000. prices ranging from 8 to 120, 180,000 to 475,000.  The Thompson—Okanagan Region had  given an average of 76,  with supply ranging from  The Kootenay Region had prices vary from 20 to 75, given an  average of 48, with supply varying from 55,000 to 175,000. From the 150 simulation runs, regional prices and their associated quantity levels  were  obtained.  Regression  between these prices and quantities. regressed  on  technique.  all  of  the  regional  analysis  used to  draw  quantities  using  a  linear  Chilcotin Region.  OLS  regression  with R—squared values ranging from  The t—stat values  for the Cariboo—  (in brackets) also appear reasonable, with  values greater than two for all but two cases.  Given the significance of the t—  the Thompson—Okanagan Region appears to be one of the more significant  regions affecting prices. large exporter of forage, price,  relationship  The regression equations are shown below.  for predicting price in the Peace River Region to 0.84  stats,  a  Note that each regional price series was  The results appear quite reasonable, 0.59  was  ‘QPe’  This is not surprising since it is a considerably on the average.  Note:  ‘P’  to production in the Peace River Region;  refers to the region’s ‘QCe’  production in the  Central Interior; ‘QCa’ production in the Cariboo—Chilcotin; ‘QTh’ production in  46 the Thompson—Okanagan;  and ‘QKo’ production in the Kootenay.  Peace River P=322—0 0005*QPe_0 0002QCe—0 0002QCa0 0001kQTh0 .0000 9*QK0 (14.6) (3.0) (4.9) (5.6) (2.4) =0 73 2 R .  .  .  .  .  Central Interior P=294—0 0002*QPe_0 0003QCe0 0003QCa0 .0002 *QTh_0 000l*QK0 (7.8) (13.0) (10.6) (5.0) (4.4) =0 .81 2 R .  .  .  .  Cariboo—Chilcotin P=309—0 000 1*QPe0 0003*QCe_0 0003QCa—0 QQQ3*Q_ 0002QKo (5.2) (11.2) (6.4) (16.0) (5.7) =0 .84 2 R .  .  .  .  .  Thompson—Okanagan P=291—0 0001*QPe_0 0002QCe—0 0003QCa0 0003QTh—0 0002QKo (8.7) (6.1) (2.5) (16.4) (5.2) =0 83 2 R .  .  .  .  .  .  Kootenay P=150 .4—0. 00004*QPe_0 00009QCe—0 0001QCa—0 0002QTh0 0002QKo (1.3) (3.5) (1.9) (8.4) (6.5) =0 59 2 R .  .  .  .  .  Since a linear regression technique was used in creating the results above, an interpretation of the parameter values is relatively meaningless as actual value changes are represented.  A more useful interpretation could be made if the  parameter values represented percentage changes (or elasticities)  .  Therefore, the  regressions were re—done using a log—linear form, since the associated parameter values would represent percentage changes. The following equations show the results of the log—linear regressions used in relating regional prices to all of the regional quantities.  These are the  equations that the Ministry of Agriculture would use in predicting regional price levels.  The following variables are defined as  %dQPE = the percentage deviation between actual and average production in the Peace River Region %dQCE the percentage deviation between actual and average production in the Central Interior Region =  %dQCA = the percentage deviation between actual and average production in the Cariboo—Chilcotin Region  47 %dQTH = the percentage deviation between actual and average production in the Thompson—Okanagan Region %dQKO = the percentage deviation between actual and average production in the Kootenay Region Also, let ‘%dPPE’ denote average regional price in the  the percentage deviation between actual and Peace River Region; similar price deviation  variables 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%dQKO .  .  .  .  .  The previous equations can be interpreted as the effect that percentage changes in all regional production has on the percentage change in each regional For example,  price.  if production was 10 percent below normal in all regions,  then price would be 22.2 percent above normal in the Peace River Region.  This  value is calculated by multiplying —10% by the parameter associated with each region in the equation.  Summing these values gives the cumulative effect of all  regional production levels being 10 percent below normal on the price in the Peace River Region.  Similarly,  a  10 percent  shortfall  in production in all  regions would lead to a price rise of 34.4 percent above normal in the Central Interior, 25.4 percent above normal in the Cariboo—Chilcotin, 32.6 percent above normal in the Thompson—Okanagan,  and 22.7 percent above normal in the Kootenay  regions. As described at the end of Chapter 4, curves was not known with certainty.  the slopes of the regional demand  The slopes of the demand curves, especially  the upper ones play an integral role in determining the levels of the predicted prices, and this in turn can have an impact on the size and frequency of payouts under the WSD insurance scheme. was  performed  on  the  upper  It is for this reason that sensitivity analysis  slopes  of  these  regional  demand  curves.  This  sensitivity analysis was done by increasing and decreasing the upper slopes of  48 the  curves  Scenario’,  in the  scenarios  labelled  ‘Steep Slope  Scenario’  and  ‘Flat  Slope  and re—doing the log—linear regressions to compare the effects on  price 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 ‘Base Case Scenario’, for each region) Scenario’  This scenario was labelled as the ‘Steep Slope  .  It was assumed that by increasing the slopes, higher price estimates  .  would 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 yields larger  price  estimates  responsive).  For  the  (i.e., ‘Steep  sloped  steeper Slope  demand  Scenario’,  the  curves result  are of  more 10  a  price  percent  shortfall in production below normal yielded an increase in regional prices above normal by 25.1 for the Peace River, Cariboo—Chilcotin, regions.  33.8  40.0 for the Central Interior, 24.6 for the  for the Thompson—Okanagan,  and 29.5  for the Kootenay  For all of these cases except for .he Cariboo—Chilcctin Region,  change in price was greater than that under the ‘Base Case Scenario’  .  the  For this  region, the ‘Steep Slope Scenario’ yielded a one percent lower increase in price than  the  ‘Base  Case  Scenario’,  which  could  probably  inaccuracy in parameterizing the regional demand curves, in  the  allocation process  (i.e.,  slightly more  forage  be  attributed  to  an  causing a slight bias 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 and the violation in the Cariboo—Chilcotin price is negligible in size. In the third scenario, the upper slopes of the regional demand curves was decreased by 50 units from the ‘Base Case Scenario’,  for each region.  assumed that this would result in smaller price estimates (i.e., demand curves are less price responsive)  .  It was  smaller sloped  See Appendix 2 for details.  A comparison in results between the ‘Flat Slope’ and ‘Base Case’ scenarios confirms the hypothesis of the lower slope being less price responsive.  For the  49 ‘Flat Slope Scenario’, a 10 percent shortfall in production below normal for all regions yields an increase in price above normal of 20.3 percent for the Peace River,  percent  29  Chilcotin, regions.  for  the  Central  Interior,  18.3  25.3 percent for the Thompson—Okanagan, For all of these cases,  percent  for  and 14.2  for the Kootenay  the  Cariboo—  the increase in price was less than for the  ‘Base Case Scenario’ The pricing model appears to respond reasonably well to changes in the slopes of the demand curves.  Larger price estimates resulted from an increase  in the slope in four of the five regions, confirming that steeper slopes are more responsive; and, lower price estimates for all regions resulted when slopes were lowered, showing that lower slopes are less price responsive.  Next, the accuracy  of the price estimates was checked. Using  the  linear  regression  results  from  the  ‘Base  Case  Scenario’,  predicted price estimates were obtained by substituting the regional quantity levels from the 150 simulations back into each regression equation.  A graphical  display showing the accuracy of the price estimation is found below in Figures 5.6 through 5.10.  These graphs represent the predicted price over the actual  price for each of the 150 simulations. approximated regional price.  A value of one represents a perfectly  50 1.4  1.2. B  B  U •UII  •  I  •  .• B  1  1--I  B  B  B 1B 1  I  .  B  B..B  I  I  :•  B  B  B  •  I  —  I  •  •  B  08-  B  S  B  B  B  0.2  -  0-  o  2.5  140  120 100 80 60 Region Predicted over real price for Peace River  40 Figure 5.6  20  -  B  2  0.  a  S  1.5  a  B  I  •  •  S  a  B  B  •if .  a  Sal. I  . 1 .5IhSSS a. • Id,B . . a S• a •  •  ••  dl  B I 5  S  •‘ .  . B  a. I S  0.5  -  -  B  0-  I  I  0  20  40 Figure 5.7  —I--  I  120 100 80 60 Predicted over re& price for Central Interior Region  I  140  —  51  1.8 1.6 1.4  -  S  a  $ &1.2--. •  “  1 •  Q  •  a. S  a  .  •  ‘h.  •  •  •  • a  .  •  .?  Ia  I.  a  a  .5  0  0.4 0.2  160  140  120 100 80 60 40 Figure 5.8 Predicted over real price for Cariboo-Chilcotin Region  20  0  I  —r  I  I  0  1.6 1.4 a  S  $1.2’ a  a  SB  I  a  SB  •a. 0.8-.  0.4  • •  •1.IS  •  a  aiIIiIa  1.5  1.  •  •aa• a •  a..  a  •  5%  •  -  1.  0.2-  I  0— 0  20  I  I  I  120 100 80 60 40 Region kanagan for price Thompson-O real over Predicted Figure 5.9  140  160  52  2.5  -  I  2-  8 I  1.5 U U  U  40 Figure 5.10  20  0  I  I  I  0  120 100 80 60 Region Kootenay Predicted over real price for  The results appear quite reasonable, prices converging to their actual values average for the regions,  160  140  with all of the estimated regional  (converging around the value one)  .  On  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 in  the four of the five regions over the 150 simulations, with Peace River being the excluded region. The  Central  Interior  Region  exhibited  two  cases  in  which  price  was  underestimated, with the more significant estimate being approximately 20 percent of actual price.  Only two or three overestimates were found, with the largest  being greater than two times the actual price.  These extreme estimates are most  likely 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 other trading regions’ prices, and there is a significant jump from the upper to lower curves,  situations can possibly arise in which the price may appear overly high  53 or low.  An example of this would be where a region produces excess production  and its surrounding regions have a significant excess demand. excess  As a result the  supply will flow to the surrounding regions such that the region will  consume at its base requirement and have price generated from the other regions prices.  A jump in that region’s price level would be observed,  as compared to  if the regions had a strictly linear demand curves (no inelastic portion)  .  Since  the regressions used are based on a linear fit, the results may appear to be an extreme over or underestimation in price. From a policy perspective, the implications of an overestimation in price are more severe than an underestimation.  If price is severely overestimated, and  the observation falls into the category in which a Wide—Spread Drought payment must be made (i.e., the estimated price exceeding the price trigger level and the actual  supply  falls  below  the  guarantee  level),  then  it  can  become  quite  expensive for the agency supplying the compensation. The Cariboo—Chilcotin Region only had one notable outlier in its price estimates, percent.  where the estimated price exceeded the actual by approximately The  Thompson—Okanagan Region had  one  extreme  underestimate,  60  where  estimated price was approximately 20 percent of the actual and two overestimates of approximately 50 percent over the actual. most number of outliers.  The Kootenay Region exhibited the  In over approximately five cases, the estimated price  exceeded the actual by more than 50 percent with the largest being more than 100 percent.  This  can  be  explained  by  the  fact  that  the  Kootenay  Region  is  significantly isolated from the rest of the province, given the huge transport costs especially beyond the Thompson—Okanagan Region.  As a result the price in  the Kootenay Region may have relatively larger fluctuations in price than the other regions. 32  31  An observed overestimation may result if the region has excess supply like in the example above, 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 does almost all of its trade with the Thompson-Okanagan Region; therefore, the Kootenay Region may  54 As stated previously, the purpose of this thesis was to develop a pricing model for forage.  With this model,  estimates of forage prices can be made and  applied directly to the problem of crop insurance.  Forage prices are a necessary  component in a WSD insurance scheme since they are used to determine the level of payments, world.  and estimates are needed as prices are non—observable in the real  The way in which this model would be applied to the WSD insurance scheme  will be described. Given the results from the regressions of prices on quantities  (assuming  that the ‘Base Case Scenario’ parameters were used), the regression equations can be used to obtain current price estimates.  The Ministry of Agriculture  (the  insurers) would substitute the current regional production levels for forage back into each of the regression equations to obtain regional price estimates.  Given  these regional price estimates and their associated quantity levels, the insurers could determine the indemnities (if any) owed to the farmers under the insurance scheme.  As described previously,  this scheme,  for a region to qualify for a payment under  actual supply must fall below the guarantee level and price must  exceed a price trigger level. The  following graph,  Figure  5.11,  payments under the WSD insurance scheme.  provides  a  graphical  description of  The results from the 150 simulations  under the ‘Base Case Scenario’ for the Peace River Region are shown in the plots of  predicted  guarantee coverage  price  34 level (amount  are  on  supply  levels.  represented.  below the  Payment  guarantee  (amount above the price trigger)  .  Note  that  levels  level)  the  price  trigger  and  are equal to the quantity  multiplied by the price  covered  The points in the top left quadrant represent  Wide—Spread Drought payments, where supply falls below the guarantee level and the 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.  55  160 140 120  1100  o  60 40 20 0 50000  0  100000 Figure 5.11  300000  250000  200000  150000  Regional Supply Peace River Region (Base Case Scenario)  The results from this graph are useful in that the insurers can determine the frequency and cost of the program.  In 150 simulations  (years), the number  of times in which a payment is made to the Peace River Region can be ascertained by the  number of observations  in the top left  quadrant.  The  cost  of these  payments can be determined by the amount by which the observation exceeds the price trigger and falls below the guarantee level. and level  of payments are of significant  Note that both the frequency  importance to the viability of the  program.  Both the frequency and levels will change as the price trigger and guarantee levels are changed.  56  Chapter 6 : Conclusions  During times of wide—spread drought,  forage producers and consumers,  in  particular those in British Columbia, may experience drastic increases in forage prices.  It had been proposed that a method be devised in which farmers could  insure themselves against this price—risk associated with these types of natural disasters.  The Wide—Spread Drought insurance scheme was proposed by the Crop  Insurance Branch Foods  at British Columbia Ministry of Agriculture,  in order to deal with this  issue.  This insurance  Fisheries,  and  scheme did however,  require the use of a model to estimate forage prices based on regional supplies of forage, developed  since prices were non—observable. a model  to  estimate  The first portion of this thesis  forage prices.  The  second portion  showed  an  application to British Columbia forage production, with a description of its use in a Wide—Spread Drought insurance scheme. Aside  from  the  huge  complexities of the model,  limitation  on  availability  of  data  and  some reasonable results were obtained.  vast  Under the  ‘Base Case Scenario’ regression analysis was used to form a relationship between changes in regional production levels and changes in regional price levels. the  regressions,  reasonable.  the  The  R—squared  accuracy of  values  the model  and in  t—statistics  all  estimating prices  graphically representing predicted prices over real prices.  appeared was  From quite  checked by  With the exception  of a few extreme values, the model worked quite well in price estimation. Results from two other sloped scenarios were compared with those of the Base Case in order to determine the models sensitivity to changes in the upper slopes of the regional demand curves. that  the changes  Log—linear regressions were performed so  in quantities and the effects on the price  expressed in percentages.  levels  could be  It was assumed that increasing the slopes would make  the model more price sensitive and lowering the slopes,  less price sensitive.  The results validated this assumption as the ‘Steep Slope Scenario’ exhibited a greater  change  in  regional  price  estimates  above  normal  levels  given  an  equivalent percentage shortfall in production below normal levels than the ‘Base  57 Case  Scenario’.  Region,  Note  that  this  true  was  for  all but  the  Cariboo—Chilcotin  in which price was only one percent less responsive than in the ‘Base  Case Scenario’  As well, the ‘Flat Slope Scenario’  .  exhibited a smaller change  in regional price estimates above normal levels given an equivalent percentage shortfall in production below normal levels than the ‘Base Case Scenario’ The  model  applicability insurance. well  in  to  developed real  in  the  thesis  world problems,  in  shows  considerable  particular to  promise  the  in  problem of  its crop  When applied to British Columbia forage production, it did reasonably  estimating forage prices  and responded quite well to  slopes of the regional demand curves.  changes  in the  However, the accuracy of these results is  questionable as some of the necessary data was unavailable.  In particular, there  forage consumption or price data available to parameterize the demand  was no curves.  As a result, approximations to these curves had to be made.  There was  no series data available on forage yields to obtain variance/covariance values for the regions, therefore, these values were obtained by drawing relationships between variability in precipitation and variability in yields.  There was also  limited availability of data on average regional yield levels, amount of seeded acreage used, In purposes,  and feed requirements per type of livestock.  order for the all  information, hypothetical.  of any  the  results  data  to ultimately be useful  requirements  conclusions  drawn  must be met.  regarding  the  for price estimation Without  results  may  this  complete  be  strictly  58  Bibliography Agriculture Canada  (1986),  Sheep Production and Marketing,  Ottawa.  Ardeni, P.C. (1989), “Does the Law of One Price Really Hold for Commodity Prices?”, American Journal of Agricultural Economics, August, p.661—669. Aumack,  Ken. (Williams Lake), Jim Forbes (Dawson Creek), Denise McLean (Vanderhoof), Mike Momberg (100 Mile House), and Ted Moore (Kamloops) Average regional forage price were estimates obtained from the aforementioned forage specialists, 1994.  Baffes, J. (1991), “Some Further Evidence on the Law of One Price : The Law of One Price Still Holds”, American Journal of Agricultural Economics, November, P.1264—73. Beames, R.M. (Dept. of Animal Science, USC), Ron Charles, Bob France, Wayne Hadow, and Marc Robbins (British Columbia Ministry of Agriculture, Fisheries, and Foods) Personal consultation was made with the aforementioned agriculture specialists, 1994. .  Beckmann, M.J. (1985), “Competitive Mill and Uniform Pricing”, Spatial Pricing and Differentiated Markets, p.52—64. Blake,  Martin J., and Tom Clevenger for Forecasting Alfalfa Ray Economics, 9(1), . 199 — 195 p.  in Norman,  G.,  (1984), “A Linked Annual and Monthly Model Prices”, Western Journal of Agricultural  Blank, S.C., and H.W. Ayer (1987), “Government Policy Cross Effects : The Cotton and Dairy Programs’ Influence of Alfalfa Hay Markets”, Agribusiness (3), p.385—392. British Columbia Ministry of Agriculture, Fisheries, and Foods, Crop Insurance Branch. Personal consultation and information on yields. British Columbia Ministry of Agriculture, Storage Options”, Victoria. Brook,  Fisheries,  and  Foods  (1994),  Anthony., David Kendrick, and Alexander Meeraus (1988), Cams Guide. Redwood City, California : The Scientific Press.  “Hay  : A User’s  California Alfalfa Hay Processing, Transporting and Pricing. (1976), Division of Agricultural Sciences, University of California, Davis. Harris, Curtis C., and Mehrzad Nadji Theory and in Practice. Lanham,  (1987), Regional Economic Modelling MD : University Press of America.  Judge,  G. George, et al. (1988), Introduction to Econometrics. Toronto : John Wiley and Sons,  Keay,  Roger C. 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Yanagida (1981), “A Methodology of Combining Annual Econometric Forecasts with Quarterly ARIMA Forecasts”, Contributed Paper,  Western Agricultural July 19—21. Meyer,  Economics  Association Meeting,  Lincoln,  Nebraska,  G.L., and J.F. Yanagida (1984), “Combining Annual Econometric Forecasts with Quarterly ARIMA Forecasts : A Heuristic Approach”, Western Journal of  Agricultural Economics,  9(1), p. . 2 — 200 06  Ministry of Tourism (1987), British Columbia Road Map and Parks Guide, Victoria. National Research Council  (1989),  “Nutrient Requirements of Domestic Animals”,  United States. Ross, C.V. (1989), Sheep—Production and Management. Englewood Cliffs, New Jersey Prentice Hall. Soder,  E.M.  (1976),  “Field Loss Aspects”,  Alfalfa Seminar, Kamloops,  B.C.  Statistics Canada (1987—1991), #22—002 Seasonal, November Estimates of Principal Field 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 Columbia 1991 Census of Agriculture, Ottawa. Takayama, T., and W.C. Labys (1986), “Spatial Equilibrium Analysis”, Handbook of Regional and Urban Economics, Vol. 1, p. 171—199. White, Kenneth J. (1993), Shazam : Econometrics Computer Program 7.0, Reference Manual. Montreal : McGraw—Hill, p.392.  User’s  Willet, K. (1983), “Single— and Multi—Commodity Models of Spatial Equilibrium in Linear Programming Framework”, a Canadian Journal of Agricultural  Economics,  July, p. . 2 — 205 22  60  Appendix 1  C)  —J  NJ  U]  U]  C)  C)  H’ C)  NJ C) C) -  —i  NJ  cc, ,.  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Di H 01  ID Di  —010  HCD 01 H  H  001 -Cl) H 0 cD  H  H-Ui H—• CD Di A  HDi  •CD  CJi -C) Di C)C)C1  C)  H 01  NJ  —  C)] C)  NJ  H 01  -  NJ NJ  —  Ui  -  H  C) H  C)  —J  I  C)C)  DiZi ODi DiO  DiO  OF-)  C)  —  Di <  C)  Cr Cl)  0  0  01 C)  NJ NJ  H Ui  NJ 01  C)  a  —  00 C)Di H-H HH0b 00 to H-I C)  C) H  01  NJ 01  H Ui  NJ  C,  OH II  0  C)CD CrC) CDrr IF-I H-Di  H  FIH  (DCI-  c-I-C)  HO C)CD  H-CD <Di 00 I-lCD  C)  H0  CD  CD  C)  Di  0  0  0  0 H Cr Di  Di  H H Di  H  H-  -U  —J C)  H  —  ‘  1<  Di  C)  CD  0  0  C)C)  DiI-tI  DcC) DiO C)  OH  C)  Cr0 H-I  C)Di H- H HH-  0 C)  II  CD  H  H  0  a  H-  Di  Di  —Cr  [)  Cl-  —  CD II  J H  CD  C)  Di  Cl)  HO  -U  CD  H  H Di  65  Appendix 2  66 Sensitivity Analysis  Log—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%dQKO .  Log—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%dQKO .  .  .  67  Appendix 3  *  GAMS  *  DEFINING THE FILE TO WRITE TO  file  FILE outt FILE con  * * *  to  run the  spatial  equilibrium model  /outt.dat/;  / 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. SET R REGIONS /PEACE CENTRAL CAR IBOO THOMPS KOOTEN STORAGE!; SET U ITERATIONS /Ul*U150/; SET Y 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) /PEACE 225 CENTRAL 154 CARIBOO 194 THOMPS 239 KOOTEN 239 STORAGE 78/;  UPPER INTERCEPT OF DEMAND CURVES  PARAMETER ALD(R) !PEACE 251 CENTRAL 180 CARIBOO 220 THOMPS 265 KOOTEN 91 STORAGE 0/;  LOWER  INTERCEPT OF DEMAND CURVES  PARAMETER BU(R) UPPER SLOPE OF DEMAND CURVES /PEACE 0.00052 CENTRAL 0.00044 CARIBOO 0.00051 THOMPS 0.00055 KOOTEN 0.0027 STORAGE 0.000189/; PARAMETER BL (R) LOWER SLOPE OF DEMAND CURVES /PEACE 0.00075 CENTRAL 0.00075 CARIBOO 0.00075 THOMPS 0.00075  KOOTEN 0.00075 STORAGE 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 216675 CENTRAL 162368 CARIBOO 208734 THOMPS 249131 KOOTEN 63461 STORAGE 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 /PEACE 195333 CENTRAL 193839 CARIBOO 111893 THOMPS 318272 KOOTEN 77925/; ALIAS *R  (R,RP);  IS ROWS AND RP  TABLE CHOL(R,RP)  PEACE CENTRAL CARIBOO THOMPS KOOTEN  PRODUCED  IS COLUMNS CHOLESKY DECOMPOSITION MATRIX FOR THE RANDOM QUANTITIES  PEACE 34539 4136 2383 5078 938  CENTRAL 0 24956 3980 6607 1033  CARIBOO 0 0 16706 6703 1059  THOMPS 0 0 0 45332 1715  KOOTEN 0 0 0 0 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  FROM SERIES  2;  STOR ADDED TO CURRENT PROD IN RUN3  * * * * * * * * *  PARAMETER 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, COM[vIAND (P.35) TO GIVE THE R SET WE HAVE TO USE THE ALIAS 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)  PEACE CENTRAL CARIBOO THOMPS KOOTEN STORAGE * *  PEACE 7 42 48 61 78 18  TRANSPORTATION COSTS BETWEEN REGIONS CENTRAL 42 13 25 43 61 21  CARIBOO 48 25 13 22 56 16  THOMPS 61 43 22 13 42 22  KOOTEN 78 61 56 42 13 21  STORAGE 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. VARIABLES 0(R) CONSUMPTION IN EACH REGION (AGENT) TR(R,RP) AMOUNT TRANSPORTED BETWEEN REGIONS W TOTAL WELFARE IA(R) INDICATOR FOR C LESS THAN CBAR 13(R) INDICATOR FOR C GREATER THAN CBAR POSITIVE 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. EQUATIONS WELFARE OBJECTIVE FUNCTION CONSUM(R) CONSUMPTION CONSTRAINTS QUANT(R) QUANTITY CONSTRAINTS STOREA(R) STORAGEA CONSTRAINT STOREB(R) STORAGEB CONSTRAINT ICONA(R) ICONSTRA CONSTRAINT ICONB(R) ICONSTRB CONSTRAINT ICONC(R) ICONSTRC CONSTRAINT ICOND(R) ICONSTRD CONSTRAINT ICONE(R) ICONSTRE CONSTRAINT; 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;  WELFARE..  7O  ICONA(R) ICONB CR).. ICONC(R).. ICONDCR).. ICONE(R).. * * *  *  * *  7,  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  *  ) =L=O; ) =G=O;  .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)  *  IA(R) * (C (R) -CB(R) 13CR) * CC CR) -CB CR) IA CR) ÷13 CR) =E=1; IA CR) =L=1; lB CR) =L=1;  HERE  /ALL/;  IS A TITLE FOR THE PRICES AND QUANTITES SENT TO outt.cIat  PUT PUT PUT PUT  outt  PUT PUT PUT PUT  con  PUT PUT PUT PUT  tostor  PUT PUT PUT PUT  storm  RESULTS  “  FROM GAMS OUTPUT!  “  QP1  “  “  P1  CONSUMPTION”  QP2  P2  “  P3  QP4  P4  QP5  STOKOO  “1;  / “1/  “  “  QP3  CONPEA  CONCEN  ALLOCATION TO  CONCAR  CONTHO  CONKOO  “I;  STORAGE’!  “1/  “  PRICESTO  “  STOPEA  STOCEN  “STORAGE ADDED TO CURRENT  STOCAR  PRODUCTION”! II  STOPEA  OPTION SEED  STOCEN =  STOTHO  STOCAR  STOTHO  // STOKOO  “/;  2576;  *CREATE THE LOOPS TO GENERATE THE PRICE DISTRIBUTION OVER SET U *********5TART OF FIRST LOOP LOOP CU, IND IND IND IND IND  C ‘PEACE’ ) =NOFMAL C 0, 1); C ‘CENTRAL’ ) =NORMAL (0, 1);  C ‘CARIBOO’ ) =NORMAL C 0, 1); C ‘THOMPS’ ) =NORMAL CD, 1); 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);  P5/;  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 Si Si Si Si  ( ( ( ( (  ‘PEACE’ , U) = TR. L ( ‘STORAGE’, ‘PEACE’); ‘CENTRAL’ ,U) = TR. L (‘STORAGE’, ‘CENTRAL’); ‘CARIBOO’ ,U) = TR.L( ‘STORAGE’, ‘CARIBOO’); ‘THOMPS’ ,U) = TR. L ( ‘STORAGE’ , ‘THOMPS’); ‘KOOTEN’ ,U) = TR. L (‘STORAGE’, ‘KOOTEN’);  ********END OF FIRST  OPTION SEED  =  LOOP  3367;  *********5TART OF LOOP  2  LOOP (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  2  OPTION SEED =2107 ******5TART LOOP 3 LOOP (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);  QP  73  ( ‘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  OPTION SEED  =  *********5TART  3  2688; LOOP  4  LOOP (U,  ( ‘PEACE’ ) =NORNAL (0, 1); ( ‘CENTRAL’ ) =NORMAL (0, 1); IND ( ‘CARIBOO’ ) =NORNAL (0, 1);  IND IND  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’ QP ( ‘KOOTEN’  ) =SUM(RP,CHOL( ‘THOMPS’ ,RP) *IND (RP) ) +M( ‘THOMPS’ ) ÷S3 (‘THOMPS’ ,U); ) =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  OPTION SEED  =  4444;  *******START LOOP LOOP (U,  4  5  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 QP  ( ‘THOMPS’ ) =STJN(RP,CHOL( ‘THOMPS’ ,RP) *IND(Rp) ) +M( ‘THOMPS’ ) +S4 (‘THOMPS’ ,U); ( ‘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  OPTION SEED  =  5  4111;  ********5TT LOOP  6  LOOP (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 QP  ( ‘THOMPS’ ) =SUM(RP, CHOL ( ‘THOMPS’ ,RP) *IND (RP) ) +M( ‘THOMPS’ ) ÷S5 ( ‘THOMPS’ ,U); ( ‘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 PRICES C.L(R)=ROUND(CL(R),4);  UU  II  (:1  111111  121121121 UUU 0-1111  HH01  i-li-li-I  UUU  1111’l(0  UUU HHH ...  -12—  O O  124 124  -I]  hI U H  r’i U H r(3  0  ci  (N  ci ci  H —  riO” •O(N cNH HH”  0  0 12) H  ‘—  (J i:I111 U  ——  HH  ••“  cici  U-  •  IXhilO HUO  --ci01  U12— 124  -  13-i”  cci H”  “O —0(0 - 13)134 1-r1H U0  ••—••  ci•o  0 H  h’IUH 13.--  Cl) 1I U  .hi1 IXH HO O riH: U---  •  ci (N H —  0 0 III H  U LI)  (l)mci “•  •..  (N(N HH  .h1 H HO O hi1 U—in  -S---  ci (Nfl H” •CN  —H  0— 0-  hflZ  Hh’I ciH c1(JQ  -111UCD  1110 124H0 --Cl)  0 H Hci (I)•ci  H(N  H11l  “H  HI U) cifl.  —-  Cl)  LflCl) ci ci.. •.C1  0  C’H  11)” 0111111 00  cD  111  Z  134 0 0 1’. 0 tflH  O1312 HO0 (l)HH •-- (0(l) 01’I—--4JU)-)  -  H.. •.-.  ci ci.. •.c’  -1 •.  ••13-Ii)  —D -Cl) -  111124  0—--’  U ..icD 11It •d13.H s-i- -  01313 -J13.HH  -)i-)  •-------  13.H - -  fi)h  O  U  H.. ..— —- Cl) 111124  (NH  U Cl)  ••ciQ ci—•• .“C’— (Ni-)H-  H.U Hci13 hi)  H”Z  (3.0’i  H  ---  —H- H - ZCl)O 1111111240 UU - 0-  —H-H  “-fl.1’IHhI U- U  -J-  (i)(l)(l)  DD 124134  0  • 0 OUU  HH DtD 13413.  JLfl  00113401134  HH DD 124134  Z12.12.l3 H  HH DD 13.13.  HH 1-TI Cl)  Z  H  1-il  01  Z  H H H  Cr..13-ci  —12.hrl 111010  H  H  111  114  H  1’)  0  •  124  0  i-)  H  •  124  H CD  13 H  -)  H C!)  z  124  —  -1313  01+  U—U--011 124 134 —i—13  H  ‘l—l’l i-)U1U •H  J}  UU —  I3.—II—II —----._-----  13  -13h±101110 U >U  00124 00 F-Il-)  *  II  Cl)  Cl) U]  i/  *  Cl) LU  *  Cl) Ui  *  Cl)  *  Cl) LU  *  Cl)  *  l>J 1’J M H H H W U] U] W i * * * * * * * ——-------—Cl) Cl) Cl) C/) Cl) Cl) Cl) M M DU H H H H  *  *-  Ii II  Wil II t-Jil II Hil II H-  CD ft hi H-  CD hi HCD i 0 CD  <  0 0  Mi  0  CD i ft U)  *W *J *H * ** * *(D H*U]**W*M*HH M) E’J lJ 1’J M CD  OD)OD.JOHOHOHOHOU1  II  MWHU1HHWPOU1ii II II II II II II II II Ii II II II  CD l)(l)CfltflCflfLj)Cl)tl)CD  CDCDCDCDCDCDCDCDCD(DCDCDCDCDCDCDCD(DCD(D(D  *  0  ‘Ij  *  -  —  Li. LI.  C)  H  Cl)  -  Zi -3  IC)  Cl)  *  —  HLi H-  H- —— *Cj) Cl)H HO C)  ti  CD H CD ft H0  0 0  Mi  0  Dl  H-  —Cl)il H H- < ——HOM—  Q—---  -—0  Q.——  *  ft hi 0 0 CD —  —-  0  H H  hi  Mi 0  hi  OH-  [-liD)  CD  H— ft— CDhhO fthi0  MCDCD CDrt OhiO H-H-CD  hi  QjCD  MiOCD MU) OCD<  H-0 >ft  <CD ft DlQH ft(D  00(0 OH(D (DCD Mi U) CD  CD —  i H-  Mi  hi Dl H-  Cl)  *  —  hi 0  —  LL  CD  —Q QQ 0ftQ H frQM CDH H H CDCDCD ft H-QjMi 0 0  0CD hi ftH QiCD DlCD ft 0 CDO(D —0  Dl ftft  QQ  •  (DCD Q  l-  H-0<  U]  LU  —  (I) i-hi-h * * * * -M.Cl) DCDH-HCDHH CD0 1’tiCDCD 1• —H CDa CDCD ft(D H00 H0 0 HHOO HP 0  *  H-  H Hft  H-  CD hi HCD  <  CD H CD ft CD  I-i  H0  0 O l-iCD OM] 1Mi H00 ftH H- CD 0 ft UI  H-CD  HH H-DJ ft <CD  H-CD  l-  DiP  *  H-  —  hi 0  *  Cl)  —.  — hi 0  -—  H-  — —  h 0  —  H-  Cl)  — H  •  WW  •  tsJH III 00  i-  *(00 (DCD  CD  <  CD (0 H0  i-  :  0  .  CD  .  MH0LI  <  LLfl  *—Q Cl)lI—  t5-—HhiM— 0CD QiH-hi —0  —LJ.  —  Cl) * H-Ci)*  *---Q  LlQ  HH- CD —ft frHCD0 H-  —Dl —h  dH0—Q  DlH  —Pl  LL-  X H-  1WD-JHhQ  H-H-l))Lfl0---  CD ft hi H——  *  fl<ftu1gU1*CD U1COHWl))Cfl OH<LJ-Cl)J--JoU]LDH-ft  H- H- hi  0 CD  *  HiI 01111 111111 O(D 0 l-N) HHPHH—i-h CD *ft * * * * *11 H()ooooo0 QCD . HH-  CDLI  <  Dl lH-  CDCDCDCD(D  *t0t  MFH-H(D —II H-<’t50  *  OCD  *  Dit1  *  0 0  Mi  0  ft U)  QHD  HCI)  W)a)LoJ—JH0  CD  CD H CI)  0 H  (Q  Dl  *—M--J0DU]DU1Wl))U1ft  ll-oM J’JJMhh llO**********HOH-00000000000 . HH-aWU](DMWU1CD  i-h  WWDJ’J’)HHHH0 WUiW’J0 Li-Oil H II II II II II II Ii CD II  H-  Q  *  IOH-0000000000LI•  *  i-  *(0(0(0( CDCDCDCDCDCDCDCDCDCD  <0  *  H  H H Ui  H  CO  —  I- -UiW) H-—  H  CD  CD  CDCO  c-c  CO  —  w H)  CO  —  P)u1u1u1u1  <COCOCOCOCO  o  0  -.LfliW)  o  H  1WD.)  CD Q.COCOCOCOCO  c1  fJwwww  I)COCOCO  CD  t’JQ HO  1h  CO  —  CDHMOrOM  II  CO  H HO  If  F-  COCOCOCOCO  H  CO  rr  L’iO 0<  —  f)COCOCO  II  krtU]WM  II  ctH bH—HH  II  OUW’)  i.<t-  ij  0  xO  COCOCOHCI2H  () CO  II  ODJ  CO  h çt H H-H-H-H-c-rH  c-c  i (D I- I- I- L-  t’i[H-  H- J (I cctrtF--ct(fl  rt  I-  *  * F-3DJ  *  OH-c1  *  cthF-3DJ  2Fd t I  f- I—  -4  II  I  II  II  CO U]  * U]  CO CO  I  W *  CO  CO  *  CO  uiU1 Ui w w Ui Ui * * *  owo woJ  COCO CO7CO  II  UiUiW  l- i-  COCOCOCO COCOCO  I  (D(D(D(D(D(D(D  *t  -4  


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