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The value of precipitation forecast information in winter wheat production
Agricultural and Forest Meteorology 95 (1999) 99±111
The value of precipitation forecast information in winter wheat production Glenn Foxa,*, Jason Turnera, Terry Gillespieb a
Department of Agricultural Economics and Business, University of Guelph, Guelph, Ont., Canada N1G 2W1 b Department of Land Resource Science, University of Guelph, Guelph, Ont., Canada N1G 2W1 Received 15 October 1998; accepted 15 November 1998
Abstract Many aspects of agricultural production can be adversely affected by weather. Weather forecast services tailored for the speci®c needs of the farming community are available throughout North America. Estimating the value of these services to farmers is increasingly important as weather service budgets are under increasing scrutiny. A framework to characterize the value of precipitation forecast information to winter wheat producers in the province of Ontario, Canada, is developed. A mean±variance model is used as the basis for this framework. This theoretical framework is applied to precipitation forecast data from the Windsor and the London weather of®ces for the crop years of 1994 and 1995. Four forecast methods are compared. A naive forecast based on precipitation over the last four days is used as the baseline forecast. The second forecast considered is the daily Environment Canada farm forecast. A third forecast was constructed by arbitrarily improving the accuracy of the Environment Canada forecast by 50%. The fourth forecast considered assumed perfect foresight on the part of producers, in the sense of knowing the actual pattern of precipitation over the next 4 days. Precipitation damage relationships during harvest are developed based on available agronomic data. The value of weather forecast information was found to vary considerably between 1994 and 1995. The level of risk aversion of the producer was also found to be an important determinant of the value of weather forecast information, although some of our results indicate that the value of weather forecast information may be inversely related to the degree of risk aversion. Estimates of the value of precipitation forecast information averaged $100.00 (CDN)/ha per year. # 1999 Elsevier Science B.V. All rights reserved. Keywords: Precipitation; Forecast; Mean±variance
1. Introduction Weather forecast information is an important management tool for crop producers. Traditionally, weather forecast information has been provided to Canadian farmers at no charge or for a cost that has been considerably less than the cost of providing this *Corresponding author. Tel.: +1-519-824-4120; fax: +1-519767-1510.
service. Budget problems of both federal and provincial levels of government in recent years have precipitated a series of changes in the terms at which services are being provided by government agencies in Canada. Estimating the value of these services to the people who use them is problematic since market transactions are usually not available to provide an indication of how much consumers of the service are willing to pay for it or what the costs of supplying the service in a competitive market would be. Valuing
0168-1923/99/$ ± see front matter # 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 1 9 2 3 ( 9 9 ) 0 0 0 2 2 - 2
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information services is particularly dif®cult. Howard et al. (1996) have recently surveyed the conceptual and empirical issues involved in estimating the value of information as well as the incentive problems associated with the production and distribution of information as a commodity. This paper presents an analytical framework to characterize the value of weather forecast information in agricultural production. The approach is based on Antonovitz and Roe (1982, 1986) and Roe and Nygaard (1980). This framework is used to estimate the value of precipitation forecast information to winter wheat producers in Southern Ontario. We analyse producers harvest timing decisions under different types of weather forecast information. Precipitation is an important factor in determining drying costs and grain quality in winter wheat in Ontario. Precipitation forecast information helps farmers choose a harvest date that optimizes drying costs and sprouting losses. The framework is based on a mean±variance risk programming model (Robison and Barry, 1987). Two general approaches to characterizing the value of information have been used in the empirical literature. The ®rst approach is a forward looking or ex ante approach. This approach measures how much better off a decision maker, in this context a farmer, would expect to be if production choices could be based on better information. The second approach is a backward looking or ex post approach. This approach asks how much better off the decision maker would have been if the outcome of the uncertain event about which information was needed had been known when the decision had to be made. In the ®rst approach, the estimated value of information may be subject to second thoughts if it turns out to be less accurate or relevant than the decision maker anticipated. The second approach has been criticised as 20±20 hindsight. For an ongoing information service that has a known reputation, however, there should be a relationship between values of information that are obtained with the two methods. This study uses the ex post approach. Examples of studies that have reported estimates of the value of weather forecast information include Lave (1963); Tice and Clouser (1982); Babcock (1990); Baquet et al. (1976); Brown et al., 1986; Carlson (1989); Gandin et al. (1992); Morzuch and Willis (1982); Hashemi and Decker (1969); Anaman and Lellyett (1996); Vining et al. (1984); Wilks
(1992); Stewart et al. (1984) and Warren and Leduc (1982). Recent surveys of this literature have been conducted by Murphy (1994); Anaman et al. (1995) and Johnson and Holt (1977). Economic models are based on the assumption that a producer is attempting to achieve some goal. A farmer makes decisions for his enterprise to attain this goal. Two goals have been widely used; pro®t maximization and expected utility maximization. Pro®t maximization is a useful tool for modelling farmer's decisions under conditions of certainty. This model assumes that the farmer knows all prices, yields and costs with certainty when producing a commodity. However, this is not very useful when modelling farmers decisions under risky conditions. It is important to develop an objective function that can incorporate how a farmer makes decisions in a risky environment. The theory of expected utility maximization provides a foundation for which models of this type can be developed. 2. The expected utility model Bernoulli was the ®rst to propose that people maximize expected utility rather than expected monetary payoff. Utility is the level of satisfaction that a person experiences in a speci®c set of circumstances. The expected utility model assumes that when people face uncertainty about what circumstances will occur, they weigh each potential outcome by the probability of its occurrence. Suppose that U(a) is the utility of a person when situation `a' occurs and U(b) the utility that the same person would experience if situation `b' occurs. If `a' and `b' are the only possible outcomes, and if the probability that `a' will occur is p, then the expected utility is pU(a) (1 ÿ p)U(b). Expected utility indicates the level of satisfaction that a person expects to experience when that person faces uncertainty. The expected utility model assumes that people choose the option that gives the highest expected utility. Bernoulli used this proposition to explain why people were willing to pay below $20 for a gamble with an expected value of in®nity (Schoemaker, 1982). Bernoulli studied the St. Petersburg paradox. The St. Petersburg paradox is based on a game in which the player tosses a fair coin repeatedly. The player receives a payoff equal to $2n, where n is the number
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of coin tosses that the player makes before `tails' appear. The expected value of a player's winnings in a single try at this game is in®nite. But most people are only willing to pay a few dollars for the opportunity to play such a game. This observation led Bernoulli to conclude that people do not evaluate uncertain situations on the basis of expected values alone. Risk preferences also matter. Hence, the expected utility model has been proposed. (See Machina (1987) for an assessment of the expected utility model.) VonNeumann and Morgenstern demonstrated that expected utility maximization was a rational decision criterion. If a decision maker obeys speci®c axioms, a utility function can be formulated that re¯ects the preferences of the decision maker. (Schoemaker, 1982; Robison and Barry, 1987) The expected utility model assumes that people evaluate alternatives, in the face of uncertain information, according to the expected utility of each choice or action. This can be written as X Xj max f pi U si where f(pi) represents the probability of state i and U(Xj /Si) the utility incurred from state Si when action Xj is taken. Expected utility is summed over the set of possible outcomes or states that can occur with nonzero probabilities. Risk preference is re¯ected in the curvature of the utility function with respect to the outcome argument. If the outcome argument of the utility function is expressed as, say, pro®t, aversion to risk is manifest as concavity of the utility function in pro®t. Concavity indicates a declining marginal utility of pro®t. Risk aversion implies that individuals must be compensated for taking risks, in the form of a premium, over and above the return on a completely certain action (Robison and Barry, 1987). The difference between the expected return on a risky action and the return on a riskless action which leaves the farmer indifferent between the two choices is de®ned as the risk premium. Determination of a risk premium is illustrated in Fig. 1. A concave utility function for a risk averse individual is presented. An individual can take a risky action with possible outcomes, 1 and 2, each with a given probability. The expected value of these outcomes is E(). The expected utility from these out-
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Fig. 1. Risk premium and certainty equivalent profit.
comes is EU(). Alternatively, the individual could accept a certain outcome CE. The utility associated with this outcome, U(CE), is equal to the utility the individual expects to receive from the risky action, EU(). The return on the risk-free action is equal to the expected return on the risky action, less the risk premium, R. It is de®ned as the certainty equivalent of the expected return on the risky action: Certainty equivalent expected risky return ÿrisk premium For risk averse decision makers, the risk premium is positive to provide compensation needed for risk bearing (Robison and Barry, 1987). Individuals have different risk preferences and attitudes towards different levels of risk. The degree of risk-aversion impacts on decisions being made by an individual. The expected utility model is a useful tool for modelling decisions involving risky choices (Schoemaker, 1982). The general expected utility maximization framework can illustrate that different farmers incorporate weather forecast information into production decisions in different ways. Some farmers may prefer to use weather forecast information on their farm to reduce the risk that weather events have on the production of commodities. Other farmers may not use weather forecast information because they are willing to accept the risk of not knowing what future weather events are in store. This model captures the concept that each person reacts to risk differently and has different preferences.
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The mean±variance pro®t model used in this study is a special case of the more general expected utility model. It assumes that the decision maker exhibits constant absolute risk aversion and faces normally distributed risks. Constant absolute risk aversion means that the intensity of the decision makers risk preference does not change as pro®t or wealth changes. (See Robison and Barry (1987)) for a more complete discussion. The mean±variance pro®t model has been widely used in risk analysis in agricultural production. Although it is less general than the expected utility model, it is much less cumbersome to apply. 2.1. The mean±variance profit model The mean±variance pro®t model characterizes expected utility maximization problems in terms of the expected value and the variance of pro®t (Robison and Barry, 1987). Expected value-variance sets describe a risk-ef®cient frontier. The risk-ef®cient frontier is de®ned as the r set of choices that provide minimum variance for alternative levels of expected returns. Fig. 2 illustrates an ef®cient frontier labelled AB. Choices along curve AB are preferred to choices below curve AB. For each point below curve AB there is a point on the ef®cient set that has a higher expected pro®t for the same variance or the same expected pro®t with a lower variance. Choices C and D have the same expected value although choice C has a lower variance. Therefore, choice C is preferred to choice D.
Fig. 2. The mean±variance frontier.
Points above frontier AB cannot be attained so the area above frontier AB is known as the infeasible region. Robison and Barry (1987) illustrate how the mean± variance pro®t model can be used to characterize the value of information. The objective function of the model is: max CE E ÿ 2 2 This objective function is maximized subject to the feasible set of expected pro®t and variance of pro®t, where CE is certainty equivalent or mean±variance pro®t, E() the expected pro®t, the Arrow±Pratt coef®cient of absolute risk aversion and 2 is the variance of pro®t. The Arrow±Pratt coef®cient of risk aversion is a measure of the intensity of risk preference. It is de®ned as ÿU 00 y=U 0 y. A positive value for indicates risk aversion. The larger the magnitude of the greater the intensity of that risk aversion. This objective function is linear in expected pro®t and variance of pro®t. Maximization of this objective function is equivalent to ®nding an optimal solution based on the equality of slope between the ef®cient set and the indifference curves describing income and risk preferences (Robison and Barry, 1987). Fig. 3 illustrates how differences in intensity of risk preference are re¯ected in the certainty equivalent pro®t model. The straight lines in the ®gure indicate combinations of expected income and variance of income that give equal expected utility. The more risk averse the decision maker is, the more steeply sloped are the lines in expected pro®t/variance of pro®t space. Increases in
Fig. 3. Indifference curves with different levels of risk aversion.
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the expected value-variance frontier represents the mean±variance pro®t maximizing combination of expected pro®t, P1, and variance of pro®t, V1, subject to the feasible region for a producer. The slope of this line is /2, where is the Arrow±Pratt absolute risk aversion coef®cient. Line U1 represents a more risk averse individual. The tangency point is at a lower variance of pro®t V0 and a lower expected pro®t P0. The new certainty equivalent pro®t is represented in the diagram as CE1. 2.2. The mean±variance profit model and weather forecast information Fig. 4. Alternative solutions to the mean±variance model with differnt levels of risk aversion.
expected income represent a gain in utility whereas increases in the variance in income decrease utility, so the indifference curve passing through point A in the diagram indicates a lower level of utility than the indifference curve passing through point B. In general, movements to the top or to the left within the axes of this Figure indicate increases in expected utility. Fig. 4 illustrates how the feasible frontier and the mean±variance objective function are used to identify an optimal choice for a risk averse decision maker. Line U0 describes the indifference curve between expected returns and the variance of those returns for a producer with a speci®c Arrow±Pratt coef®cient of absolute risk aversion. Certainty equivalent pro®t is labelled CE0 in the diagram. Tangency of line U0 and
Fig. 5 illustrates how the mean±variance pro®t model can be used to determine the value of weather information. The horizontal axis represents the level of some production decision, X. This could be the level of some productive input to apply, the level of output selected, or, in the present context, the timing of harvest. The vertical axis represents the value of the mean±variance objective function per hectare obtained from that production decision. Mean±variance pro®t with perfect forecast information is labelled ex post pro®t. Because there is no variance associated with this relationship, pro®t and meanvariance pro®t are the same. The best choice of X, in terms of ex post pro®t, is labelled X*. The relationship that the farmer believes to hold between this production action and mean-variance pro®t is labelled anticipated mean±variance pro®t. A producer maximizing his anticipated mean±variance pro®t would
Fig. 5. The value of weather forecast information.
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choose X0. But the pro®t that he will realize, having chosen X0, is determined by the ex post pro®t relationship. The height of the ex post pro®t relationship at X0 is what the farmer will actually earn. The difference between this value and the height of the ex post pro®t relationship at X* is the value of, in this case, perfect information about whatever factor or factors caused the uncertainty in the ®rst place. Alternatively, the farmer could obtain additional information before he makes his production decision. Armed with this additional information, his beliefs about the mean±variance pro®t that he faces would change. If he acts on this new information and selects a different X he will realize pro®ts indicated by the height of the ex post pro®t relationship at that level of X. The difference between this value of ex post pro®ts at this new level of X and the corresponding value at X0 is the value of the better information.
Environment Canada forecast, the improved forecast and the perfect forecast.
2.3. Forecast method
Weather forecast information is an important management tool in deciding when to harvest winter wheat in Ontario. Winter wheat is planted in the fall in Ontario and is harvested the following July. Growers can time the harvest of the crop based on weather forecast information. Signi®cant damage to each crop can occur if production activities are not performed at the appropriate time. Weather forecast information is useful for timing planting, spraying, fertilizing and harvesting of winter wheat. Precipitation forecast information is most valuable at harvest time. Grain quality deteriorates when rain delays wheat harvest (Bauer and Black, 1983b). Losses in grain quality result from sprouting and a reduction in test weight. Relationships developed for the impact of rainfall on sprouting and grain moisture content are taken from a study looking at hard red spring wheat (Bauer and Black, 1983a). Winter wheat grown in Ontario is soft white wheat. This wheat is more sensitive to rainfall than the hard wheat and may sprout more easily. The model used in this study de®nes quality by the percentage of grain sprouted at harvest. In Ontario, quality is determined not only by the amount of sprouting but also by the test weight of the grain. This was not included in the model as a relationship of rainfall and test weight was not determined. This also would have increased complexity of the model dramatically. For this study, grain quality is de®ned by the percentage amount of grain sprouted at harvest. Rain-
Four forecast methods were considered in this study, including a naive forecast, the actual Environment Canada weather forecast, an improved Environment Canada weather forecast and a perfect weather forecast. The naive forecast used actual weather conditions from the current day as the forecast for the next day. Actual weather conditions from the past 2, 3 and 4 days are used for forecasts, for forecast days 2, 3 and 4. The Environment Canada forecast was the farm weather forecast issued at 6 : 30 a.m. The improved forecast arbitrarily enhanced the accuracy of the forecast provided by Environment Canada by 50%. Suppose Environment Canada forecast the most likely amount of precipitation to be 2 mm for a given day. On this day, the actual precipitation amount was 4 mm. The improved forecast for this day would, therefore, be 3 mm of precipitation. The perfect forecast used the actual weather conditions for a given day as the forecast values for the same day. There is no error associated with this forecast method. Mean±variance pro®t was calculated under each forecast method and optimal harvest timing was determined under each forecast method. Pro®t realized by the producer for the actual weather conditions that occurred for each harvest timing decision was calculated. Actual pro®t under the naive forecast method is used as the benchmark to determine the value of the
2.4. Risk preferences Different values of the Arrow±Pratt coef®cient are used to examine the impact that the intensity of risk preference has on the value of weather forecast information. Preckel et al. (1987) reported coef®cients ranging from 0.0000021 to 0.000037. For this study Arrow-Pratt coef®cient of absolute risk aversion values of 0.01, 0.001, 0.0001 and 0.00001 are used to test the sensitivity of optimal harvest dates and the value of weather forecast information to variations in the degree of risk aversion of producers. 3. The winter wheat model
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fall also increases grain moisture content. Drying charges are incurred by the producer if grain moisture content is >14.5%. Weather forecast and actual weather occurrence data for 1994 and 1995 are used for this study. Data from the Windsor and London weather of®ces are used for the winter wheat model. These areas account for a large proportion of winter wheat produced in the province. 3.1. Calculation of forecast variance The precipitation forecast can be interpreted as an example of a truncated normal distribution. A truncated normal distribution is illustrated in Fig. 6. Amount of forecast precipitation, R, measured in millimetres, is on the horizontal axis and marginal probability for each level of forecast precipitation occurring, f(R), is on the vertical axis. The distribution is truncated at 0 mm of precipitation as it is not possible to have negative precipitation. The mean±variance pro®t model requires that we know the mean and the variance of this truncated normal distribution. Forecasts issued by Environment Canada are used to develop a distribution for each day's forecast. The area under the distribution represents the probability of precipitation forecast by Environment Canada, which is 60% in the example illustrated in Fig. 6. The amount of precipitation, M, represents the most likely amount of precipitation
105
forecast by Environment Canada. This amount is the expected value of precipitation for the truncated distribution. It is possible to calculate the variance of the truncated normal distribution using the probability of precipitation, the most likely amount of precipitation and the fact that the distribution is normal and truncated at 0 mm. Forecast variance is calculated for each day's forecast using formulas from Johnson and Kotz (1970) for mean and variance of a truncated distribution. Forecast most likely amount of precipitation variance is then used for calculating variance of pro®t. Each day's forecast probability of precipitation and most likely amount of precipitation is assumed to be independent of all other forecasts. The variance for each of the 4 days included in the forecast is added together to obtain total forecast variance for each 4day period. For numerical reasons, forecast precipitation amounts of 0 mm are replaced by 0.1 mm, forecast probabilities of 0% are input as 1% probabilities of precipitation and forecast probabilities of 100% are input as 99% probabilities of precipitation to calculate forecast variance. 3.2. Harvest decision The wheat producer's harvest decision is modelled as follows. On each day during the potential harvest period, mean±variance pro®t is calculated if the producer harvests on that day and also for harvest on each of the following 3 days. If the current day exhibits the highest mean±variance pro®t for the next 4-day period, the farmer will harvest the crop on that day. If one of the following 3 days has a higher mean±variance pro®t, the farmer waits until the next day. On the next day, based on the weather forecast information that becomes available on that day, mean±variance pro®ts for the next 4-day period are calculated. If, on the ®rst day of this second 4-day period, the mean±variance pro®t is higher on the ®rst day, harvest occurs on that day. Otherwise, the farmer waits one more day and recalculates the mean±variance pro®ts again. This continues until the farmer has selected a day to harvest the crop. 3.3. Grain yield and rainfall damage
Fig. 6. Truncated normal distribution of forecast precipitation.
Winter wheat yield is determined by weather conditions over the entire crop growing period. Producers
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can do little to change ®nal crop yield. Average crop yields for the 1994 and 1995 growing season are used in the model. Wheat harvests for Essex and Middlesex counties are modelled. Average wheat prices in Ontario for 1994 and 1995 were $138 and $193/tonne, respectively. Average yield for 1994 and 1995 was 4.25 tonnes/ha. These counties account for a large percentage of winter wheat production in southern Ontario. Crop damage at harvest is modelled as sprouting and high grain moisture content. Studies into the impact of rainfall on sprouting have been conducted (Bauer and Black, 1983a). Results of a study by Bauer and Black (1983a) showed for increased rainfall amounts of 0, 4 and 8 mm, sprouting of 0.25% per day/mm occurred. Based on these results, sprouting, S, measured as percentage grain yield sprouted, is calculated as: S 0:25R where R is the accumulated rainfall after wheat maturity date, measured in millimetres. Sprouting percentage increases as accumulated rainfall amount increases. Final crop yield is impacted negatively by sprouting. Final yield, Y, takes into consideration amount of sprouting in the grain and is calculated as: Y YI 1 ÿ S where YI is the initial grain yield and S the percentage sprouts in the grain for a given day. In Ontario, wheat grain harvested with a moisture content higher than 14.5% is subject to drying charges. Wheat harvested prior to maturity will be at a higher moisture content than required. Wheat that has been rained on will also be at a higher moisture content. Wheat grain begins to dry when the rain ends. Bauer and Black (1983b) present results of the impact of rainfall on grain moisture. A relationship based on these results for wheat moisture content on day t, Mt, is calculated as: Mt 0:25Mtÿ1 4:2Pt 9:375 where Mtÿ1 is the grain moisture content from the previous day and Pt is the current days forecast rainfall. Grain moisture content and sprouting are calculated for each day in the forecast.
3.4. Expected profit Pro®t, , is calculated as the product of wheat price, P, and wheat yield, Y, less drying costs, c: PY ÿ c where P is measured in dollars per tonne, Y in metric tons per hectare and c in dollars per hectare. Based on interviews with local elevators, drying costs per metric ton of winter wheat are described by ct 1:24 1:80 Mt ÿ 14:5 A ®xed charge of $1.24 is levied on each metric ton of wheat to be dried and an additional cost of $1.80 per metric ton is applied for each 1% of moisture >14.5%. Taking drying charges, ct and sprouting, S, into account, pro®t is calculated as: PYI 1 ÿ S ÿ 1:8YI Mt ÿ 14:5 12:4 where pro®t is measured in dollars per hectare. Rainfall amount is a random variable, so the sprouting percentage and the moisture level are random variables. Expected pro®t on forecast day t, E(t), measured in dollars per hectare is calculated as: E t PYI 1ÿE St ÿ1:8Y E Mt ÿ14:512:4 where E(St) is the expected sprouting and E(Mt) the expected grain moisture content on day t. 3.5. Variance of profit Sprouting and moisture content depend not only on rainfall on the forecast day but also on forecast rainfall amounts on the previous forecast dates included in the 4-day forecast. For example, variance of pro®t for the fourth day of the 4-day forecast consists of forecast variance from forecast Day 1, Day 2, Day 3 and Day 4. Because of this, variance of pro®t on Day 1 of the forecast is calculated differently than variance of pro®t on Day 4 of the forecast period. Forecast variance for each day is calculated using the truncated distribution described above. Variance of pro®t, 2(), measured in dollars per hectare, is calculated using the equations for pro®t and expected pro®t. Based on the relationships de®ned above, the variance of pro®t for forecast Day 1, 2(1) is calculated as 2 1 0:25PYI 2 2R1 ÿ 7:56YI 2 2R1
G. Fox et al. / Agricultural and Forest Meteorology 95 (1999) 99±111
where 2R1 is forecast most likely amount of precipitation variance on forecast Day 1 measured in millimetres. Variance of pro®t on forecast Day 2, 2(2) is calculated as 2
2
2
0:25PYI 2R1 2:79YI 2 2R1
2R2
ÿ
7:56YI 2 2R2
where 2R2 is forecast most likely amount of precipitation variance on forecast Day 2 measured in millimetres. Variance of pro®t on forecast Day 3, 2(3) is calculated as 2 3 0:25PYI 2 2R1 2R2 2R3 ÿ 7:56YI 2 2R3 2:79YI 2 2R2 1:03YI 2 2R1 where 2R3 is forecast most likely amount of precipitation variance on forecast Day 3. Variance of pro®t on forecast Day 4, 2(4) is calculated as 2 4 0:25PYI 2 2R1 2R2 2R3 2R4 ÿ 7:56YI 2 2R4 2:79YI 2 2R3 1:03YI 2 2R2 0:38YI 2 2R1 where 2R4 is forecast most likely amount of precipitation variance on forecast Day 4. 3.6. Certainty equivalent profit Mean±variance pro®t, measured in dollars per hectare for the wheat model is calculated for each of the four forecast days. Mean±variance pro®t for forecast Day 1, (CE,1), is calculated as CE;1 PYI 1 ÿ E S1 ÿ c2 ÿ 2 0:25PYI 2 2R1 7:56YI 2 2R1
where E(S1) is expected sprouting on forecast Day 1 and c1 the drying cost on forecast Day 1. Certainty equivalent pro®t for forecast Day 2, (CE,2), is calculated as CE;2 PYI 1 ÿ E S2 ÿ c2 ÿ 2 0:25PYI
2R1 2R2 7:56YI 2 2R2 2:79YI 2 2R1
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where E(S2) is expected sprouting on forecast Day 2 and c2 is drying cost on forecast Day 2. Certainty equivalent pro®t for forecast Day 3, (CE,3), is calculated as CE;3 PYI 1 ÿ E S3 ÿ c3 ÿ 2 0:25PYI 2 2R1 2R2 2R3 7:56YI 2 2R3 2:79YI 2 2R2 1:03YI 2 2R1 where E(S3) is expected sprouting on forecast Day 3 and c3 is drying cost on forecast Day 3. Certainty equivalent pro®t for forecast Day 4, (CE,4) is calculated as CE;4 PYI 1 ÿ E S4 ÿ c4 ÿ 2 0:25PYI 2
2R1 2R2 2R3 2R4 7:56YI 2 2R4 2:79YI 2
where E(S4) is expected sprouting on forecast Day 4 and c4 is drying cost on forecast Day 4. Winter wheat harvest dates that maximize certainty equivalent pro®t for each precipitation forecast are calculated. The value of precipitation forecast information is calculated as a difference in pro®ts between following a naive forecast and one of the other forecasts. Pro®t per hectare that is realized when the certainty equivalent pro®t maximizing decision for the naive forecast is followed and the actual weather for the harvest period occurs is subtracted from the realized pro®t arising when one of the other forecast methods is used. For example, the value of the Environment Canada forecast, VE is determined by calculating the difference in realized pro®t from following the Environment Canada forecast, WE, and the realized pro®t from following the naive forecast, WN: VE WE ÿ WN The value of the improved forecast method, VI is determined by calculating the difference in realized pro®t per hectare with the improved forecast, WI, with the naive forecast, WN: VI WI ÿ WN Finally, the value of a perfect forecast, VP, is calculated as the difference in realized pro®t with a perfect forecast, WP, and with a naive forecast, WN: VP WP ÿ WN
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Table 1 Optimal 1994 and 1995 harvest dates for different forecast methods and Arrow±Pratt coefficients for Windsor weather station Forecast method
Optimal harvest date
Expected sprouting (%)
Actual sprouting (%)
Expected moisture content (%)
Actual moisture content (%)
1994 Naive ( 0.01, 0.001, 0.0001, 0.0001) Environment Canada ( 0.01, 0.001) Environment Canada ( 0.0001, 0.00001) Improved ( 0.01) Improved ( 0.001, 0.0001, 0.00001) Perfect ( 0.01, 0.001, 0.0001, 0.0001)
24 25 27 26 27 27
July July July July July July
0
0
20.5 18.7 14.25 19.51 14.25 14.25
20.5 40.54 14.25 19.51 14.25 14.25
1995 Naive ( 0.01, 0.001, 0.0001, 0.0001) Environment Canada ( 0.01, 0.001, 0.0001, 0.0001) Improved ( 0.01, 0.001, 0.0001, 0.0001) Perfect ( 0.01, 0.001, 0.0001, 0.0001)
24 30 30 30
July July July July
0 2.95 2.95 2.95
2.95 2.95 2.95 2.95
20.5 13.56 13.56 13.56
64.18 13.56 13.56 13.56
for 1994 and 1995. For example, for the Windsor weather of®ce, in 1994, the optimal harvest date using the naive forecast method was 24 July with expected sprouting of 0% and expected moisture content of 20.5%. Actual sprouting and moisture content were also 0 and 20.5% for harvest on 24 July. In this context, `actual' means estimated moisture content or sprouting based on observed precipitation. The optimal harvest date for the Environment Canada, improved and perfect forecast methods fell between 25 and 27 July, depending on the degree of risk aversion. In
Values are measured in dollars per hectare. Forecast values are calculated for weather forecasts from the Windsor weather of®ce and the London weather of®ce. Calculations are made for the 1994 and 1995 growing seasons. 3.7. Optimal harvest dates Optimal harvest dates for each forecast method were calculated for the Windsor weather of®ce (Table 1) and the London weather of®ce (Table 2)
Table 2 Optimal 1994 and 1995 harvest dates for different forecast methods and Arrow±Pratt coefficients for London weather station Forecast method
Optimal harvest date
Expected sprouting (%)
Actual sprouting (%)
Expected moisture content (%)
Actual moisture content (%)
1994 Naive ( 0.01, 0.001, 0.0001, 0.0001) Environment Canada ( 0.01) Environment Canada ( 0.001, 0.0001, 0.00001) Improved ( 0.01) Improved ( 0.001, 0.0001, 0.0001) Perfect ( 0.01, 0.001, 0.0001, 0.0001)
24 25 27 26 27 27
July July July July July July
0 0 0.05 0.025 0.05 0.05
0 0 0.05 0.05 0.05 0.05
20.5 24.79 14.11 18.51 14.11 14.11
40.86 34.87 14.11 18.93 14.11 14.11
1995 Naive ( 0.01, 0.001, 0.0001, 0.0001) Naive ( 0.0001, 0.0001) Environment Canada ( 0.01, 0.001, 0.0001, 0.0001) Improved ( 0.01, 0.001, 0.0001, 0.0001) Perfect ( 0.01, 0.001, 0.0001, 0.0001)
25 27 27 27 27
July July July July July
0 0.2 0.1 0.125 0.15
0 0.15 0.15 0.15 0.15
22.90 15.15 13.47 13.89 14.31
21.22 14.31 14.31 14.31 14.31
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Table 3 Value of different forecast methods for winter wheat harvest for varying degrees of risk preference at Windsor weather station for 1994 and 1995 Forecast Method
Environment Canada Improved Perfect
1994
1995
0.01
0.001
0.0001 and 0.00001
0.01, 0.001, 0.0001 and 0.00001
$(153.31)/ha $7.57/ha $47.14/ha
$(153.31)/ha $47.14/ha $47.14/ha
$47.14/ha $47.14/ha $47.14/ha
$363.99/ha $363.99/ha $363.99/ha
1995, the optimal harvest date using the naive forecast method was 24 July with expected sprouting of 0% and expected moisture content of 20.5%. Actual sprouting on this date was 2.95% and actual moisture content as 64.18%. Optimal harvest date for the Environment Canada, improved and perfect forecast methods was found to be 30 July with expected and actual sprouting of 2.95% and moisture content of 13.56%. The optimal harvest date for the London weather of®ce using the naive forecast method in 1995 was sensitive to the risk preference of the producer (Table 2). A producer with an Arrow±Pratt coef®cient of absolute risk aversion of 0.01 or 0.001 had an optimal harvest date of 25 July. A producer with an Arrow±Pratt coef®cient of absolute risk aversion of 0.0001 or 0.00001 had an optimal harvest date of 27 July. 4. Forecast value Estimates of the value of precipitation forecast information for 1994 and 1995 for the Windsor and London weather of®ce are presented in Tables 3 and 4. The values reported indicate the increase in realized pro®ts per hectare relative to a harvest decision based on a naive forecast. For the Windsor weather station
(Table 3) in 1994, the Environment Canada, improved and perfect forecast methods all had a value of $47.14/ ha for a producer with an Arrow±Pratt coef®cient of absolute risk aversion of 0.0001 or 0.00001. All of these combinations of levels of risk aversion and types of forecast resulted in an optimal harvest date of 27 July, which was the best date ex post. For higher levels of risk aversion of 0.001 and 0.01, the Environment Canada forecast actually did worse than the naive forecast. At these values of the Arrow±Pratt coef®cient of absolute risk aversion, the optimal harvest date for the Environment Canada forecast was 25 July. But selecting this harvest date turned out, ex post, to be a mistake and rain would have lead to a dramatic increase in moisture content (see Table 1, Line 2). For 1995, the Environment Canada, improved and perfect forecast methods all had a value of $363.99/ha for all Arrow±Pratt coef®cient values that we considered. For the London station (Table 4), the naive forecast indicated an optimal harvest date of 24 July. The optimal harvest date with the highest level of risk aversion with the Environment Canada forecast was 25 July. For more moderate levels of risk aversion, the optimal harvest date for the Environment Canada forecast was 27 July, which turned out to be the best harvest date ex post. As a result, the Environment Canada forecast was only worth $76.42/ha to the most risk averse producer while it was worth $233.20 to producers
Table 4 Value of different forecast methods for winter wheat harvest for varying degrees of risk preference at London weather station for 1994 and 1995 Forecast method
Environment Canada Improved Perfect
1994
1995
0.01
0.001, 0.0001 and 0.00001
0.01 and 0.001
0.0001 and 0.00001
$76.42/ha $198.05/ha $233.20/ha
$233.20/ha $233.20/ha $233.20/ha
$51.77/ha $51.77/ha $51.77/ha
$0/ha $0/ha $0/ha
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Table 5 Average value of weather forecast information for varying degrees of producer risk preference Forcast method
0.01
0.001
0.0001 and 0.00001
Environment Canada Improved Perfect
$84.72/ha $155.35/ha $174.03/ha
$110.97/ha $161.08/ha $161.08/ha
$161.08/ha $161.08/ha $161.08/ha
with less extreme risk preferences. The improved and perfect forecast method also had a value of $233.20/ha in 1994 for Arrow±Pratt coef®cient values of 0.001, 0.0001 and 0.00001. In 1995, in the London area, the Environment Canada improved and perfect forecast methods each had a value of $51.77/ha for a producer with an Arrow±Pratt coef®cient of absolute risk aversion of 0.01 or 0.001. For less risk averse producers, (0.0001 or 0.00001) the Environment Canada, improved and perfect forecast methods had no value relative to a naive forecast since the naive forecast indicated an optimal harvest date of 27 July for these levels of risk aversion and that turned out to be the best harvest date ex post. The London area results are an example of one of the paradoxes sometimes encountered in the value of information. Better forecast information was worth more to less risk averse producers in 1994 but was worth less to those same producers in 1995. But as Hilton (1981) has demonstrated, our intuition that the value of information should vary with the degree of risk aversion is not necessarily true. The effect of variations in the Arrow±Pratt coef®cient of absolute risk aversion is reported in Table 5. These results indicate the value of the Environment Canada precipitation forecast to wheat producers is inversely related to degree of risk aversion. This situation arose because the model chose earlier harvest dates for higher levels of risk aversion. This higher degree of risk aversion made producers more vulnerable to the effects of an inaccurate forecast in the two crop years in question. 5. Discussion Harvested area of winter wheat in the province of Ontario has been in the neighbourhood of 300 000 ha in recent years. According to the results obtained with
the model presented in this paper, the value of Environment Canada's precipitation forecast varied from a value of negative $153.31/ha for very risk averse producers in the Windsor area in 1994 (Table 3) to a positive value of $363.99/ha for producers in the same area in 1995 (Table 3). The average of this range is still over $100.00/ha per year. The average of the range of values computed for producers in the London area was also over $100.00/ha per year. This amounts to about one-®fth of the gross revenue per hectare typically obtained from winter wheat. If these values are applied to the typical harvested area of 3 00 000 hectares per year, this would amount to an aggregate value of weather forecast information in winter wheat harvesting of about $30 million annually. But our ®ndings indicate that there is considerable variation in the value of precipitation forecast information to winter wheat producers across years and for different levels of risk aversion. This study has focussed on the characterization of the value of the bene®ts of precipitation forecast information in agricultural production. Although it is outside the scope of this study, we wish to acknowledge the importance a complementary analysis that, as far as we know, has not been undertaken. This is the estimation of the costs of providing this forecast information. It is, after all, the value of the bene®ts relative to the costs of this service that interests the cost-bene®t analyst. Acknowledgements Research support of the Ontario Ministry of Agriculture, Food and Rural Affairs and the Atmospheric Environment Service of Environment Canada is gratefully acknowledged. Comments by Dr. K. T. Paw U and the two reviewers are gratefully acknowledged. References Anaman, K., Thampapillai, D., Henderson-Sellers, A., Noar, P., Sullivan, P., 1995. Methods for assessing the benefits of meteorological services in Australia. Meteorological Applications 2(1), 17±29. Anaman, K., Lellyett, S., 1996. Assessment of the benefits of an enhanced weather information services for the cotton industry in Australia. Meteorological Applications 3(2), 127±135.
G. Fox et al. / Agricultural and Forest Meteorology 95 (1999) 99±111 Antonovitz, F., Roe, T., 1982. A measure of the value of information for the competitive firm under price uncertainty. American Agricultural Economics Association Meeting. pp. 1± 16. Antonovitz, F., Roe, T., 1986. A theoretical and empirical approach to the value of information in risky markets. Review Econ. Stat. 68, 105±114. Babcock, B., 1990. The value of weather information in market equilibrium. Am. J. Agric. Econ. 72(1), 63±72. Baquet, A., Halter, A., Conklin, F., 1976. The value of frost forecasting: a bayesian appraisal. Am. J. Agric. Econ. 58(1), 511±519. Bauer, A., Black, A., 1983a. Sprouting in intact mature spikes of hard red spring wheat. Agron. J. 75(6), 1016±1022. Bauer, A., Black, A., 1983b. Rain-induced harvest losses in swathed and standing wheat. North Dakota Research Report 97, 3±11. Brown, B., Katz, R., Murphy, A., 1986. On the economic value of seasonal precipitation forecasts: the fallowing/planting problem. Bull. Am. Meteorological Soc. 67(7), 833±841. Carlson, J., 1989. The importance of agricultural weather information: a Michigan survey. Bull. Am. Meteorological Soc. 70(4), 366±380. Gandin, L., Murphy, A., Zhukovsky, E., 1992. Economically Optimal Decisions and the Value of Meteorological Information, pp. 64±71. Hashemi, F., Decker, W., 1969. Using climatic information and weather forecasts for decisions in economizing irrigation water. Agric. Meteorol. 6, 245±257. Hilton, R., 1981. The determinants of information value: synthesizing some general results. Manage. Sci. 27(1), 57±64. Howard, W., Fox, G., Turvey, C., 1996. The economic benefits of new information technology. Working Paper 3/96, Policy Branch. Agriculture and Agrifood Canada, Ottawa. Johnson, S., Holt, M., 1977. The value of weather information. In: Katz, R., Murphy, A. (Eds.). Economic Value of Weather and Climate Forcasts. Cambridge University Press, Cambridge.
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Johnson, N.L., Kotz, S., 1970. Continuous Univariate Distributions. Houghton±Mifflin, New York. Lave, L., 1963. The value of better weather information to the raisin industry. Econometrica 31(1±2), 51±164. Machina, M., 1987. Choice under uncertainty: problems solved and unsolved. J. Economic Perspectives 1(1), 121±154. Morzuch, B., Willis, C., 1982. Value of weather information in cranberry marketing decisions. Bull. Am. Meteorological Soc. 21, 499±504. Murphy, A., 1994. Assessing the economic value of weather forecasts: an overview of methods, results and issues. Meteorological Applications 1, 69±73. Preckel, P., Loehman, E., Kaylen, M., 1987. The value of public information for microeconomic production decisions. Western J. Agric. Econ. 12(2), 193±197. Robison, L., Barry, P., 1987. The Competitive Firms Response to Risk. MacMillan, New York. Roe, T., Nygaard, D., 1980. Wheat, allocative error and risk: Northern Tunisia. University of Minnesota Economic Development Center Bulletin, University of Minnesota, St. Paul. Schoemaker, P., 1982. The expected utility model: its variants, purposes, evidence and limitations. J. Econ. Lit. 20(2), 529± 563. Stewart, T., Katz, R., Murphy, A., 1984. Value of weather information. Bull. Am. Meteorological Soc. 65(2), 126±137. Tice, T., Clouser, R., 1982. Determination of the value of weather information to individual corn producers. J. Appl. Meteorol. 21(4), 447±452. Vining, K., Pope, C., Dugas, W., 1984. Usefulness of weather information to Texas agricultural producers. Bull. Am. Meteorol. Soc. 65(12), 1316±1319. Warren, H., Leduc, S., 1982. An empirical valuation of meteorological information. J. Appl. Meteorol. 21, 1455±1465. Wilks, D., 1992. Tactical decision making using short-range forecasts: timing alfalfa harvests. Atmospheric Environment Services: Fifth International Meeting on Statistical Climatology.
The value of precipitation forecast information in winter wheat production Glenn Foxa,*, Jason Turnera, Terry Gillespieb a
Department of Agricultural Economics and Business, University of Guelph, Guelph, Ont., Canada N1G 2W1 b Department of Land Resource Science, University of Guelph, Guelph, Ont., Canada N1G 2W1 Received 15 October 1998; accepted 15 November 1998
Abstract Many aspects of agricultural production can be adversely affected by weather. Weather forecast services tailored for the speci®c needs of the farming community are available throughout North America. Estimating the value of these services to farmers is increasingly important as weather service budgets are under increasing scrutiny. A framework to characterize the value of precipitation forecast information to winter wheat producers in the province of Ontario, Canada, is developed. A mean±variance model is used as the basis for this framework. This theoretical framework is applied to precipitation forecast data from the Windsor and the London weather of®ces for the crop years of 1994 and 1995. Four forecast methods are compared. A naive forecast based on precipitation over the last four days is used as the baseline forecast. The second forecast considered is the daily Environment Canada farm forecast. A third forecast was constructed by arbitrarily improving the accuracy of the Environment Canada forecast by 50%. The fourth forecast considered assumed perfect foresight on the part of producers, in the sense of knowing the actual pattern of precipitation over the next 4 days. Precipitation damage relationships during harvest are developed based on available agronomic data. The value of weather forecast information was found to vary considerably between 1994 and 1995. The level of risk aversion of the producer was also found to be an important determinant of the value of weather forecast information, although some of our results indicate that the value of weather forecast information may be inversely related to the degree of risk aversion. Estimates of the value of precipitation forecast information averaged $100.00 (CDN)/ha per year. # 1999 Elsevier Science B.V. All rights reserved. Keywords: Precipitation; Forecast; Mean±variance
1. Introduction Weather forecast information is an important management tool for crop producers. Traditionally, weather forecast information has been provided to Canadian farmers at no charge or for a cost that has been considerably less than the cost of providing this *Corresponding author. Tel.: +1-519-824-4120; fax: +1-519767-1510.
service. Budget problems of both federal and provincial levels of government in recent years have precipitated a series of changes in the terms at which services are being provided by government agencies in Canada. Estimating the value of these services to the people who use them is problematic since market transactions are usually not available to provide an indication of how much consumers of the service are willing to pay for it or what the costs of supplying the service in a competitive market would be. Valuing
0168-1923/99/$ ± see front matter # 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 1 9 2 3 ( 9 9 ) 0 0 0 2 2 - 2
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information services is particularly dif®cult. Howard et al. (1996) have recently surveyed the conceptual and empirical issues involved in estimating the value of information as well as the incentive problems associated with the production and distribution of information as a commodity. This paper presents an analytical framework to characterize the value of weather forecast information in agricultural production. The approach is based on Antonovitz and Roe (1982, 1986) and Roe and Nygaard (1980). This framework is used to estimate the value of precipitation forecast information to winter wheat producers in Southern Ontario. We analyse producers harvest timing decisions under different types of weather forecast information. Precipitation is an important factor in determining drying costs and grain quality in winter wheat in Ontario. Precipitation forecast information helps farmers choose a harvest date that optimizes drying costs and sprouting losses. The framework is based on a mean±variance risk programming model (Robison and Barry, 1987). Two general approaches to characterizing the value of information have been used in the empirical literature. The ®rst approach is a forward looking or ex ante approach. This approach measures how much better off a decision maker, in this context a farmer, would expect to be if production choices could be based on better information. The second approach is a backward looking or ex post approach. This approach asks how much better off the decision maker would have been if the outcome of the uncertain event about which information was needed had been known when the decision had to be made. In the ®rst approach, the estimated value of information may be subject to second thoughts if it turns out to be less accurate or relevant than the decision maker anticipated. The second approach has been criticised as 20±20 hindsight. For an ongoing information service that has a known reputation, however, there should be a relationship between values of information that are obtained with the two methods. This study uses the ex post approach. Examples of studies that have reported estimates of the value of weather forecast information include Lave (1963); Tice and Clouser (1982); Babcock (1990); Baquet et al. (1976); Brown et al., 1986; Carlson (1989); Gandin et al. (1992); Morzuch and Willis (1982); Hashemi and Decker (1969); Anaman and Lellyett (1996); Vining et al. (1984); Wilks
(1992); Stewart et al. (1984) and Warren and Leduc (1982). Recent surveys of this literature have been conducted by Murphy (1994); Anaman et al. (1995) and Johnson and Holt (1977). Economic models are based on the assumption that a producer is attempting to achieve some goal. A farmer makes decisions for his enterprise to attain this goal. Two goals have been widely used; pro®t maximization and expected utility maximization. Pro®t maximization is a useful tool for modelling farmer's decisions under conditions of certainty. This model assumes that the farmer knows all prices, yields and costs with certainty when producing a commodity. However, this is not very useful when modelling farmers decisions under risky conditions. It is important to develop an objective function that can incorporate how a farmer makes decisions in a risky environment. The theory of expected utility maximization provides a foundation for which models of this type can be developed. 2. The expected utility model Bernoulli was the ®rst to propose that people maximize expected utility rather than expected monetary payoff. Utility is the level of satisfaction that a person experiences in a speci®c set of circumstances. The expected utility model assumes that when people face uncertainty about what circumstances will occur, they weigh each potential outcome by the probability of its occurrence. Suppose that U(a) is the utility of a person when situation `a' occurs and U(b) the utility that the same person would experience if situation `b' occurs. If `a' and `b' are the only possible outcomes, and if the probability that `a' will occur is p, then the expected utility is pU(a) (1 ÿ p)U(b). Expected utility indicates the level of satisfaction that a person expects to experience when that person faces uncertainty. The expected utility model assumes that people choose the option that gives the highest expected utility. Bernoulli used this proposition to explain why people were willing to pay below $20 for a gamble with an expected value of in®nity (Schoemaker, 1982). Bernoulli studied the St. Petersburg paradox. The St. Petersburg paradox is based on a game in which the player tosses a fair coin repeatedly. The player receives a payoff equal to $2n, where n is the number
G. Fox et al. / Agricultural and Forest Meteorology 95 (1999) 99±111
of coin tosses that the player makes before `tails' appear. The expected value of a player's winnings in a single try at this game is in®nite. But most people are only willing to pay a few dollars for the opportunity to play such a game. This observation led Bernoulli to conclude that people do not evaluate uncertain situations on the basis of expected values alone. Risk preferences also matter. Hence, the expected utility model has been proposed. (See Machina (1987) for an assessment of the expected utility model.) VonNeumann and Morgenstern demonstrated that expected utility maximization was a rational decision criterion. If a decision maker obeys speci®c axioms, a utility function can be formulated that re¯ects the preferences of the decision maker. (Schoemaker, 1982; Robison and Barry, 1987) The expected utility model assumes that people evaluate alternatives, in the face of uncertain information, according to the expected utility of each choice or action. This can be written as X Xj max f pi U si where f(pi) represents the probability of state i and U(Xj /Si) the utility incurred from state Si when action Xj is taken. Expected utility is summed over the set of possible outcomes or states that can occur with nonzero probabilities. Risk preference is re¯ected in the curvature of the utility function with respect to the outcome argument. If the outcome argument of the utility function is expressed as, say, pro®t, aversion to risk is manifest as concavity of the utility function in pro®t. Concavity indicates a declining marginal utility of pro®t. Risk aversion implies that individuals must be compensated for taking risks, in the form of a premium, over and above the return on a completely certain action (Robison and Barry, 1987). The difference between the expected return on a risky action and the return on a riskless action which leaves the farmer indifferent between the two choices is de®ned as the risk premium. Determination of a risk premium is illustrated in Fig. 1. A concave utility function for a risk averse individual is presented. An individual can take a risky action with possible outcomes, 1 and 2, each with a given probability. The expected value of these outcomes is E(). The expected utility from these out-
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Fig. 1. Risk premium and certainty equivalent profit.
comes is EU(). Alternatively, the individual could accept a certain outcome CE. The utility associated with this outcome, U(CE), is equal to the utility the individual expects to receive from the risky action, EU(). The return on the risk-free action is equal to the expected return on the risky action, less the risk premium, R. It is de®ned as the certainty equivalent of the expected return on the risky action: Certainty equivalent expected risky return ÿrisk premium For risk averse decision makers, the risk premium is positive to provide compensation needed for risk bearing (Robison and Barry, 1987). Individuals have different risk preferences and attitudes towards different levels of risk. The degree of risk-aversion impacts on decisions being made by an individual. The expected utility model is a useful tool for modelling decisions involving risky choices (Schoemaker, 1982). The general expected utility maximization framework can illustrate that different farmers incorporate weather forecast information into production decisions in different ways. Some farmers may prefer to use weather forecast information on their farm to reduce the risk that weather events have on the production of commodities. Other farmers may not use weather forecast information because they are willing to accept the risk of not knowing what future weather events are in store. This model captures the concept that each person reacts to risk differently and has different preferences.
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The mean±variance pro®t model used in this study is a special case of the more general expected utility model. It assumes that the decision maker exhibits constant absolute risk aversion and faces normally distributed risks. Constant absolute risk aversion means that the intensity of the decision makers risk preference does not change as pro®t or wealth changes. (See Robison and Barry (1987)) for a more complete discussion. The mean±variance pro®t model has been widely used in risk analysis in agricultural production. Although it is less general than the expected utility model, it is much less cumbersome to apply. 2.1. The mean±variance profit model The mean±variance pro®t model characterizes expected utility maximization problems in terms of the expected value and the variance of pro®t (Robison and Barry, 1987). Expected value-variance sets describe a risk-ef®cient frontier. The risk-ef®cient frontier is de®ned as the r set of choices that provide minimum variance for alternative levels of expected returns. Fig. 2 illustrates an ef®cient frontier labelled AB. Choices along curve AB are preferred to choices below curve AB. For each point below curve AB there is a point on the ef®cient set that has a higher expected pro®t for the same variance or the same expected pro®t with a lower variance. Choices C and D have the same expected value although choice C has a lower variance. Therefore, choice C is preferred to choice D.
Fig. 2. The mean±variance frontier.
Points above frontier AB cannot be attained so the area above frontier AB is known as the infeasible region. Robison and Barry (1987) illustrate how the mean± variance pro®t model can be used to characterize the value of information. The objective function of the model is: max CE E ÿ 2 2 This objective function is maximized subject to the feasible set of expected pro®t and variance of pro®t, where CE is certainty equivalent or mean±variance pro®t, E() the expected pro®t, the Arrow±Pratt coef®cient of absolute risk aversion and 2 is the variance of pro®t. The Arrow±Pratt coef®cient of risk aversion is a measure of the intensity of risk preference. It is de®ned as ÿU 00 y=U 0 y. A positive value for indicates risk aversion. The larger the magnitude of the greater the intensity of that risk aversion. This objective function is linear in expected pro®t and variance of pro®t. Maximization of this objective function is equivalent to ®nding an optimal solution based on the equality of slope between the ef®cient set and the indifference curves describing income and risk preferences (Robison and Barry, 1987). Fig. 3 illustrates how differences in intensity of risk preference are re¯ected in the certainty equivalent pro®t model. The straight lines in the ®gure indicate combinations of expected income and variance of income that give equal expected utility. The more risk averse the decision maker is, the more steeply sloped are the lines in expected pro®t/variance of pro®t space. Increases in
Fig. 3. Indifference curves with different levels of risk aversion.
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the expected value-variance frontier represents the mean±variance pro®t maximizing combination of expected pro®t, P1, and variance of pro®t, V1, subject to the feasible region for a producer. The slope of this line is /2, where is the Arrow±Pratt absolute risk aversion coef®cient. Line U1 represents a more risk averse individual. The tangency point is at a lower variance of pro®t V0 and a lower expected pro®t P0. The new certainty equivalent pro®t is represented in the diagram as CE1. 2.2. The mean±variance profit model and weather forecast information Fig. 4. Alternative solutions to the mean±variance model with differnt levels of risk aversion.
expected income represent a gain in utility whereas increases in the variance in income decrease utility, so the indifference curve passing through point A in the diagram indicates a lower level of utility than the indifference curve passing through point B. In general, movements to the top or to the left within the axes of this Figure indicate increases in expected utility. Fig. 4 illustrates how the feasible frontier and the mean±variance objective function are used to identify an optimal choice for a risk averse decision maker. Line U0 describes the indifference curve between expected returns and the variance of those returns for a producer with a speci®c Arrow±Pratt coef®cient of absolute risk aversion. Certainty equivalent pro®t is labelled CE0 in the diagram. Tangency of line U0 and
Fig. 5 illustrates how the mean±variance pro®t model can be used to determine the value of weather information. The horizontal axis represents the level of some production decision, X. This could be the level of some productive input to apply, the level of output selected, or, in the present context, the timing of harvest. The vertical axis represents the value of the mean±variance objective function per hectare obtained from that production decision. Mean±variance pro®t with perfect forecast information is labelled ex post pro®t. Because there is no variance associated with this relationship, pro®t and meanvariance pro®t are the same. The best choice of X, in terms of ex post pro®t, is labelled X*. The relationship that the farmer believes to hold between this production action and mean-variance pro®t is labelled anticipated mean±variance pro®t. A producer maximizing his anticipated mean±variance pro®t would
Fig. 5. The value of weather forecast information.
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choose X0. But the pro®t that he will realize, having chosen X0, is determined by the ex post pro®t relationship. The height of the ex post pro®t relationship at X0 is what the farmer will actually earn. The difference between this value and the height of the ex post pro®t relationship at X* is the value of, in this case, perfect information about whatever factor or factors caused the uncertainty in the ®rst place. Alternatively, the farmer could obtain additional information before he makes his production decision. Armed with this additional information, his beliefs about the mean±variance pro®t that he faces would change. If he acts on this new information and selects a different X he will realize pro®ts indicated by the height of the ex post pro®t relationship at that level of X. The difference between this value of ex post pro®ts at this new level of X and the corresponding value at X0 is the value of the better information.
Environment Canada forecast, the improved forecast and the perfect forecast.
2.3. Forecast method
Weather forecast information is an important management tool in deciding when to harvest winter wheat in Ontario. Winter wheat is planted in the fall in Ontario and is harvested the following July. Growers can time the harvest of the crop based on weather forecast information. Signi®cant damage to each crop can occur if production activities are not performed at the appropriate time. Weather forecast information is useful for timing planting, spraying, fertilizing and harvesting of winter wheat. Precipitation forecast information is most valuable at harvest time. Grain quality deteriorates when rain delays wheat harvest (Bauer and Black, 1983b). Losses in grain quality result from sprouting and a reduction in test weight. Relationships developed for the impact of rainfall on sprouting and grain moisture content are taken from a study looking at hard red spring wheat (Bauer and Black, 1983a). Winter wheat grown in Ontario is soft white wheat. This wheat is more sensitive to rainfall than the hard wheat and may sprout more easily. The model used in this study de®nes quality by the percentage of grain sprouted at harvest. In Ontario, quality is determined not only by the amount of sprouting but also by the test weight of the grain. This was not included in the model as a relationship of rainfall and test weight was not determined. This also would have increased complexity of the model dramatically. For this study, grain quality is de®ned by the percentage amount of grain sprouted at harvest. Rain-
Four forecast methods were considered in this study, including a naive forecast, the actual Environment Canada weather forecast, an improved Environment Canada weather forecast and a perfect weather forecast. The naive forecast used actual weather conditions from the current day as the forecast for the next day. Actual weather conditions from the past 2, 3 and 4 days are used for forecasts, for forecast days 2, 3 and 4. The Environment Canada forecast was the farm weather forecast issued at 6 : 30 a.m. The improved forecast arbitrarily enhanced the accuracy of the forecast provided by Environment Canada by 50%. Suppose Environment Canada forecast the most likely amount of precipitation to be 2 mm for a given day. On this day, the actual precipitation amount was 4 mm. The improved forecast for this day would, therefore, be 3 mm of precipitation. The perfect forecast used the actual weather conditions for a given day as the forecast values for the same day. There is no error associated with this forecast method. Mean±variance pro®t was calculated under each forecast method and optimal harvest timing was determined under each forecast method. Pro®t realized by the producer for the actual weather conditions that occurred for each harvest timing decision was calculated. Actual pro®t under the naive forecast method is used as the benchmark to determine the value of the
2.4. Risk preferences Different values of the Arrow±Pratt coef®cient are used to examine the impact that the intensity of risk preference has on the value of weather forecast information. Preckel et al. (1987) reported coef®cients ranging from 0.0000021 to 0.000037. For this study Arrow-Pratt coef®cient of absolute risk aversion values of 0.01, 0.001, 0.0001 and 0.00001 are used to test the sensitivity of optimal harvest dates and the value of weather forecast information to variations in the degree of risk aversion of producers. 3. The winter wheat model
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fall also increases grain moisture content. Drying charges are incurred by the producer if grain moisture content is >14.5%. Weather forecast and actual weather occurrence data for 1994 and 1995 are used for this study. Data from the Windsor and London weather of®ces are used for the winter wheat model. These areas account for a large proportion of winter wheat produced in the province. 3.1. Calculation of forecast variance The precipitation forecast can be interpreted as an example of a truncated normal distribution. A truncated normal distribution is illustrated in Fig. 6. Amount of forecast precipitation, R, measured in millimetres, is on the horizontal axis and marginal probability for each level of forecast precipitation occurring, f(R), is on the vertical axis. The distribution is truncated at 0 mm of precipitation as it is not possible to have negative precipitation. The mean±variance pro®t model requires that we know the mean and the variance of this truncated normal distribution. Forecasts issued by Environment Canada are used to develop a distribution for each day's forecast. The area under the distribution represents the probability of precipitation forecast by Environment Canada, which is 60% in the example illustrated in Fig. 6. The amount of precipitation, M, represents the most likely amount of precipitation
105
forecast by Environment Canada. This amount is the expected value of precipitation for the truncated distribution. It is possible to calculate the variance of the truncated normal distribution using the probability of precipitation, the most likely amount of precipitation and the fact that the distribution is normal and truncated at 0 mm. Forecast variance is calculated for each day's forecast using formulas from Johnson and Kotz (1970) for mean and variance of a truncated distribution. Forecast most likely amount of precipitation variance is then used for calculating variance of pro®t. Each day's forecast probability of precipitation and most likely amount of precipitation is assumed to be independent of all other forecasts. The variance for each of the 4 days included in the forecast is added together to obtain total forecast variance for each 4day period. For numerical reasons, forecast precipitation amounts of 0 mm are replaced by 0.1 mm, forecast probabilities of 0% are input as 1% probabilities of precipitation and forecast probabilities of 100% are input as 99% probabilities of precipitation to calculate forecast variance. 3.2. Harvest decision The wheat producer's harvest decision is modelled as follows. On each day during the potential harvest period, mean±variance pro®t is calculated if the producer harvests on that day and also for harvest on each of the following 3 days. If the current day exhibits the highest mean±variance pro®t for the next 4-day period, the farmer will harvest the crop on that day. If one of the following 3 days has a higher mean±variance pro®t, the farmer waits until the next day. On the next day, based on the weather forecast information that becomes available on that day, mean±variance pro®ts for the next 4-day period are calculated. If, on the ®rst day of this second 4-day period, the mean±variance pro®t is higher on the ®rst day, harvest occurs on that day. Otherwise, the farmer waits one more day and recalculates the mean±variance pro®ts again. This continues until the farmer has selected a day to harvest the crop. 3.3. Grain yield and rainfall damage
Fig. 6. Truncated normal distribution of forecast precipitation.
Winter wheat yield is determined by weather conditions over the entire crop growing period. Producers
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G. Fox et al. / Agricultural and Forest Meteorology 95 (1999) 99±111
can do little to change ®nal crop yield. Average crop yields for the 1994 and 1995 growing season are used in the model. Wheat harvests for Essex and Middlesex counties are modelled. Average wheat prices in Ontario for 1994 and 1995 were $138 and $193/tonne, respectively. Average yield for 1994 and 1995 was 4.25 tonnes/ha. These counties account for a large percentage of winter wheat production in southern Ontario. Crop damage at harvest is modelled as sprouting and high grain moisture content. Studies into the impact of rainfall on sprouting have been conducted (Bauer and Black, 1983a). Results of a study by Bauer and Black (1983a) showed for increased rainfall amounts of 0, 4 and 8 mm, sprouting of 0.25% per day/mm occurred. Based on these results, sprouting, S, measured as percentage grain yield sprouted, is calculated as: S 0:25R where R is the accumulated rainfall after wheat maturity date, measured in millimetres. Sprouting percentage increases as accumulated rainfall amount increases. Final crop yield is impacted negatively by sprouting. Final yield, Y, takes into consideration amount of sprouting in the grain and is calculated as: Y YI 1 ÿ S where YI is the initial grain yield and S the percentage sprouts in the grain for a given day. In Ontario, wheat grain harvested with a moisture content higher than 14.5% is subject to drying charges. Wheat harvested prior to maturity will be at a higher moisture content than required. Wheat that has been rained on will also be at a higher moisture content. Wheat grain begins to dry when the rain ends. Bauer and Black (1983b) present results of the impact of rainfall on grain moisture. A relationship based on these results for wheat moisture content on day t, Mt, is calculated as: Mt 0:25Mtÿ1 4:2Pt 9:375 where Mtÿ1 is the grain moisture content from the previous day and Pt is the current days forecast rainfall. Grain moisture content and sprouting are calculated for each day in the forecast.
3.4. Expected profit Pro®t, , is calculated as the product of wheat price, P, and wheat yield, Y, less drying costs, c: PY ÿ c where P is measured in dollars per tonne, Y in metric tons per hectare and c in dollars per hectare. Based on interviews with local elevators, drying costs per metric ton of winter wheat are described by ct 1:24 1:80 Mt ÿ 14:5 A ®xed charge of $1.24 is levied on each metric ton of wheat to be dried and an additional cost of $1.80 per metric ton is applied for each 1% of moisture >14.5%. Taking drying charges, ct and sprouting, S, into account, pro®t is calculated as: PYI 1 ÿ S ÿ 1:8YI Mt ÿ 14:5 12:4 where pro®t is measured in dollars per hectare. Rainfall amount is a random variable, so the sprouting percentage and the moisture level are random variables. Expected pro®t on forecast day t, E(t), measured in dollars per hectare is calculated as: E t PYI 1ÿE St ÿ1:8Y E Mt ÿ14:512:4 where E(St) is the expected sprouting and E(Mt) the expected grain moisture content on day t. 3.5. Variance of profit Sprouting and moisture content depend not only on rainfall on the forecast day but also on forecast rainfall amounts on the previous forecast dates included in the 4-day forecast. For example, variance of pro®t for the fourth day of the 4-day forecast consists of forecast variance from forecast Day 1, Day 2, Day 3 and Day 4. Because of this, variance of pro®t on Day 1 of the forecast is calculated differently than variance of pro®t on Day 4 of the forecast period. Forecast variance for each day is calculated using the truncated distribution described above. Variance of pro®t, 2(), measured in dollars per hectare, is calculated using the equations for pro®t and expected pro®t. Based on the relationships de®ned above, the variance of pro®t for forecast Day 1, 2(1) is calculated as 2 1 0:25PYI 2 2R1 ÿ 7:56YI 2 2R1
G. Fox et al. / Agricultural and Forest Meteorology 95 (1999) 99±111
where 2R1 is forecast most likely amount of precipitation variance on forecast Day 1 measured in millimetres. Variance of pro®t on forecast Day 2, 2(2) is calculated as 2
2
2
0:25PYI 2R1 2:79YI 2 2R1
2R2
ÿ
7:56YI 2 2R2
where 2R2 is forecast most likely amount of precipitation variance on forecast Day 2 measured in millimetres. Variance of pro®t on forecast Day 3, 2(3) is calculated as 2 3 0:25PYI 2 2R1 2R2 2R3 ÿ 7:56YI 2 2R3 2:79YI 2 2R2 1:03YI 2 2R1 where 2R3 is forecast most likely amount of precipitation variance on forecast Day 3. Variance of pro®t on forecast Day 4, 2(4) is calculated as 2 4 0:25PYI 2 2R1 2R2 2R3 2R4 ÿ 7:56YI 2 2R4 2:79YI 2 2R3 1:03YI 2 2R2 0:38YI 2 2R1 where 2R4 is forecast most likely amount of precipitation variance on forecast Day 4. 3.6. Certainty equivalent profit Mean±variance pro®t, measured in dollars per hectare for the wheat model is calculated for each of the four forecast days. Mean±variance pro®t for forecast Day 1, (CE,1), is calculated as CE;1 PYI 1 ÿ E S1 ÿ c2 ÿ 2 0:25PYI 2 2R1 7:56YI 2 2R1
where E(S1) is expected sprouting on forecast Day 1 and c1 the drying cost on forecast Day 1. Certainty equivalent pro®t for forecast Day 2, (CE,2), is calculated as CE;2 PYI 1 ÿ E S2 ÿ c2 ÿ 2 0:25PYI
2R1 2R2 7:56YI 2 2R2 2:79YI 2 2R1
107
where E(S2) is expected sprouting on forecast Day 2 and c2 is drying cost on forecast Day 2. Certainty equivalent pro®t for forecast Day 3, (CE,3), is calculated as CE;3 PYI 1 ÿ E S3 ÿ c3 ÿ 2 0:25PYI 2 2R1 2R2 2R3 7:56YI 2 2R3 2:79YI 2 2R2 1:03YI 2 2R1 where E(S3) is expected sprouting on forecast Day 3 and c3 is drying cost on forecast Day 3. Certainty equivalent pro®t for forecast Day 4, (CE,4) is calculated as CE;4 PYI 1 ÿ E S4 ÿ c4 ÿ 2 0:25PYI 2
2R1 2R2 2R3 2R4 7:56YI 2 2R4 2:79YI 2
where E(S4) is expected sprouting on forecast Day 4 and c4 is drying cost on forecast Day 4. Winter wheat harvest dates that maximize certainty equivalent pro®t for each precipitation forecast are calculated. The value of precipitation forecast information is calculated as a difference in pro®ts between following a naive forecast and one of the other forecasts. Pro®t per hectare that is realized when the certainty equivalent pro®t maximizing decision for the naive forecast is followed and the actual weather for the harvest period occurs is subtracted from the realized pro®t arising when one of the other forecast methods is used. For example, the value of the Environment Canada forecast, VE is determined by calculating the difference in realized pro®t from following the Environment Canada forecast, WE, and the realized pro®t from following the naive forecast, WN: VE WE ÿ WN The value of the improved forecast method, VI is determined by calculating the difference in realized pro®t per hectare with the improved forecast, WI, with the naive forecast, WN: VI WI ÿ WN Finally, the value of a perfect forecast, VP, is calculated as the difference in realized pro®t with a perfect forecast, WP, and with a naive forecast, WN: VP WP ÿ WN
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Table 1 Optimal 1994 and 1995 harvest dates for different forecast methods and Arrow±Pratt coefficients for Windsor weather station Forecast method
Optimal harvest date
Expected sprouting (%)
Actual sprouting (%)
Expected moisture content (%)
Actual moisture content (%)
1994 Naive ( 0.01, 0.001, 0.0001, 0.0001) Environment Canada ( 0.01, 0.001) Environment Canada ( 0.0001, 0.00001) Improved ( 0.01) Improved ( 0.001, 0.0001, 0.00001) Perfect ( 0.01, 0.001, 0.0001, 0.0001)
24 25 27 26 27 27
July July July July July July
0
0
20.5 18.7 14.25 19.51 14.25 14.25
20.5 40.54 14.25 19.51 14.25 14.25
1995 Naive ( 0.01, 0.001, 0.0001, 0.0001) Environment Canada ( 0.01, 0.001, 0.0001, 0.0001) Improved ( 0.01, 0.001, 0.0001, 0.0001) Perfect ( 0.01, 0.001, 0.0001, 0.0001)
24 30 30 30
July July July July
0 2.95 2.95 2.95
2.95 2.95 2.95 2.95
20.5 13.56 13.56 13.56
64.18 13.56 13.56 13.56
for 1994 and 1995. For example, for the Windsor weather of®ce, in 1994, the optimal harvest date using the naive forecast method was 24 July with expected sprouting of 0% and expected moisture content of 20.5%. Actual sprouting and moisture content were also 0 and 20.5% for harvest on 24 July. In this context, `actual' means estimated moisture content or sprouting based on observed precipitation. The optimal harvest date for the Environment Canada, improved and perfect forecast methods fell between 25 and 27 July, depending on the degree of risk aversion. In
Values are measured in dollars per hectare. Forecast values are calculated for weather forecasts from the Windsor weather of®ce and the London weather of®ce. Calculations are made for the 1994 and 1995 growing seasons. 3.7. Optimal harvest dates Optimal harvest dates for each forecast method were calculated for the Windsor weather of®ce (Table 1) and the London weather of®ce (Table 2)
Table 2 Optimal 1994 and 1995 harvest dates for different forecast methods and Arrow±Pratt coefficients for London weather station Forecast method
Optimal harvest date
Expected sprouting (%)
Actual sprouting (%)
Expected moisture content (%)
Actual moisture content (%)
1994 Naive ( 0.01, 0.001, 0.0001, 0.0001) Environment Canada ( 0.01) Environment Canada ( 0.001, 0.0001, 0.00001) Improved ( 0.01) Improved ( 0.001, 0.0001, 0.0001) Perfect ( 0.01, 0.001, 0.0001, 0.0001)
24 25 27 26 27 27
July July July July July July
0 0 0.05 0.025 0.05 0.05
0 0 0.05 0.05 0.05 0.05
20.5 24.79 14.11 18.51 14.11 14.11
40.86 34.87 14.11 18.93 14.11 14.11
1995 Naive ( 0.01, 0.001, 0.0001, 0.0001) Naive ( 0.0001, 0.0001) Environment Canada ( 0.01, 0.001, 0.0001, 0.0001) Improved ( 0.01, 0.001, 0.0001, 0.0001) Perfect ( 0.01, 0.001, 0.0001, 0.0001)
25 27 27 27 27
July July July July July
0 0.2 0.1 0.125 0.15
0 0.15 0.15 0.15 0.15
22.90 15.15 13.47 13.89 14.31
21.22 14.31 14.31 14.31 14.31
G. Fox et al. / Agricultural and Forest Meteorology 95 (1999) 99±111
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Table 3 Value of different forecast methods for winter wheat harvest for varying degrees of risk preference at Windsor weather station for 1994 and 1995 Forecast Method
Environment Canada Improved Perfect
1994
1995
0.01
0.001
0.0001 and 0.00001
0.01, 0.001, 0.0001 and 0.00001
$(153.31)/ha $7.57/ha $47.14/ha
$(153.31)/ha $47.14/ha $47.14/ha
$47.14/ha $47.14/ha $47.14/ha
$363.99/ha $363.99/ha $363.99/ha
1995, the optimal harvest date using the naive forecast method was 24 July with expected sprouting of 0% and expected moisture content of 20.5%. Actual sprouting on this date was 2.95% and actual moisture content as 64.18%. Optimal harvest date for the Environment Canada, improved and perfect forecast methods was found to be 30 July with expected and actual sprouting of 2.95% and moisture content of 13.56%. The optimal harvest date for the London weather of®ce using the naive forecast method in 1995 was sensitive to the risk preference of the producer (Table 2). A producer with an Arrow±Pratt coef®cient of absolute risk aversion of 0.01 or 0.001 had an optimal harvest date of 25 July. A producer with an Arrow±Pratt coef®cient of absolute risk aversion of 0.0001 or 0.00001 had an optimal harvest date of 27 July. 4. Forecast value Estimates of the value of precipitation forecast information for 1994 and 1995 for the Windsor and London weather of®ce are presented in Tables 3 and 4. The values reported indicate the increase in realized pro®ts per hectare relative to a harvest decision based on a naive forecast. For the Windsor weather station
(Table 3) in 1994, the Environment Canada, improved and perfect forecast methods all had a value of $47.14/ ha for a producer with an Arrow±Pratt coef®cient of absolute risk aversion of 0.0001 or 0.00001. All of these combinations of levels of risk aversion and types of forecast resulted in an optimal harvest date of 27 July, which was the best date ex post. For higher levels of risk aversion of 0.001 and 0.01, the Environment Canada forecast actually did worse than the naive forecast. At these values of the Arrow±Pratt coef®cient of absolute risk aversion, the optimal harvest date for the Environment Canada forecast was 25 July. But selecting this harvest date turned out, ex post, to be a mistake and rain would have lead to a dramatic increase in moisture content (see Table 1, Line 2). For 1995, the Environment Canada, improved and perfect forecast methods all had a value of $363.99/ha for all Arrow±Pratt coef®cient values that we considered. For the London station (Table 4), the naive forecast indicated an optimal harvest date of 24 July. The optimal harvest date with the highest level of risk aversion with the Environment Canada forecast was 25 July. For more moderate levels of risk aversion, the optimal harvest date for the Environment Canada forecast was 27 July, which turned out to be the best harvest date ex post. As a result, the Environment Canada forecast was only worth $76.42/ha to the most risk averse producer while it was worth $233.20 to producers
Table 4 Value of different forecast methods for winter wheat harvest for varying degrees of risk preference at London weather station for 1994 and 1995 Forecast method
Environment Canada Improved Perfect
1994
1995
0.01
0.001, 0.0001 and 0.00001
0.01 and 0.001
0.0001 and 0.00001
$76.42/ha $198.05/ha $233.20/ha
$233.20/ha $233.20/ha $233.20/ha
$51.77/ha $51.77/ha $51.77/ha
$0/ha $0/ha $0/ha
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G. Fox et al. / Agricultural and Forest Meteorology 95 (1999) 99±111
Table 5 Average value of weather forecast information for varying degrees of producer risk preference Forcast method
0.01
0.001
0.0001 and 0.00001
Environment Canada Improved Perfect
$84.72/ha $155.35/ha $174.03/ha
$110.97/ha $161.08/ha $161.08/ha
$161.08/ha $161.08/ha $161.08/ha
with less extreme risk preferences. The improved and perfect forecast method also had a value of $233.20/ha in 1994 for Arrow±Pratt coef®cient values of 0.001, 0.0001 and 0.00001. In 1995, in the London area, the Environment Canada improved and perfect forecast methods each had a value of $51.77/ha for a producer with an Arrow±Pratt coef®cient of absolute risk aversion of 0.01 or 0.001. For less risk averse producers, (0.0001 or 0.00001) the Environment Canada, improved and perfect forecast methods had no value relative to a naive forecast since the naive forecast indicated an optimal harvest date of 27 July for these levels of risk aversion and that turned out to be the best harvest date ex post. The London area results are an example of one of the paradoxes sometimes encountered in the value of information. Better forecast information was worth more to less risk averse producers in 1994 but was worth less to those same producers in 1995. But as Hilton (1981) has demonstrated, our intuition that the value of information should vary with the degree of risk aversion is not necessarily true. The effect of variations in the Arrow±Pratt coef®cient of absolute risk aversion is reported in Table 5. These results indicate the value of the Environment Canada precipitation forecast to wheat producers is inversely related to degree of risk aversion. This situation arose because the model chose earlier harvest dates for higher levels of risk aversion. This higher degree of risk aversion made producers more vulnerable to the effects of an inaccurate forecast in the two crop years in question. 5. Discussion Harvested area of winter wheat in the province of Ontario has been in the neighbourhood of 300 000 ha in recent years. According to the results obtained with
the model presented in this paper, the value of Environment Canada's precipitation forecast varied from a value of negative $153.31/ha for very risk averse producers in the Windsor area in 1994 (Table 3) to a positive value of $363.99/ha for producers in the same area in 1995 (Table 3). The average of this range is still over $100.00/ha per year. The average of the range of values computed for producers in the London area was also over $100.00/ha per year. This amounts to about one-®fth of the gross revenue per hectare typically obtained from winter wheat. If these values are applied to the typical harvested area of 3 00 000 hectares per year, this would amount to an aggregate value of weather forecast information in winter wheat harvesting of about $30 million annually. But our ®ndings indicate that there is considerable variation in the value of precipitation forecast information to winter wheat producers across years and for different levels of risk aversion. This study has focussed on the characterization of the value of the bene®ts of precipitation forecast information in agricultural production. Although it is outside the scope of this study, we wish to acknowledge the importance a complementary analysis that, as far as we know, has not been undertaken. This is the estimation of the costs of providing this forecast information. It is, after all, the value of the bene®ts relative to the costs of this service that interests the cost-bene®t analyst. Acknowledgements Research support of the Ontario Ministry of Agriculture, Food and Rural Affairs and the Atmospheric Environment Service of Environment Canada is gratefully acknowledged. Comments by Dr. K. T. Paw U and the two reviewers are gratefully acknowledged. References Anaman, K., Thampapillai, D., Henderson-Sellers, A., Noar, P., Sullivan, P., 1995. Methods for assessing the benefits of meteorological services in Australia. Meteorological Applications 2(1), 17±29. Anaman, K., Lellyett, S., 1996. Assessment of the benefits of an enhanced weather information services for the cotton industry in Australia. Meteorological Applications 3(2), 127±135.
G. Fox et al. / Agricultural and Forest Meteorology 95 (1999) 99±111 Antonovitz, F., Roe, T., 1982. A measure of the value of information for the competitive firm under price uncertainty. American Agricultural Economics Association Meeting. pp. 1± 16. Antonovitz, F., Roe, T., 1986. A theoretical and empirical approach to the value of information in risky markets. Review Econ. Stat. 68, 105±114. Babcock, B., 1990. The value of weather information in market equilibrium. Am. J. Agric. Econ. 72(1), 63±72. Baquet, A., Halter, A., Conklin, F., 1976. The value of frost forecasting: a bayesian appraisal. Am. J. Agric. Econ. 58(1), 511±519. Bauer, A., Black, A., 1983a. Sprouting in intact mature spikes of hard red spring wheat. Agron. J. 75(6), 1016±1022. Bauer, A., Black, A., 1983b. Rain-induced harvest losses in swathed and standing wheat. North Dakota Research Report 97, 3±11. Brown, B., Katz, R., Murphy, A., 1986. On the economic value of seasonal precipitation forecasts: the fallowing/planting problem. Bull. Am. Meteorological Soc. 67(7), 833±841. Carlson, J., 1989. The importance of agricultural weather information: a Michigan survey. Bull. Am. Meteorological Soc. 70(4), 366±380. Gandin, L., Murphy, A., Zhukovsky, E., 1992. Economically Optimal Decisions and the Value of Meteorological Information, pp. 64±71. Hashemi, F., Decker, W., 1969. Using climatic information and weather forecasts for decisions in economizing irrigation water. Agric. Meteorol. 6, 245±257. Hilton, R., 1981. The determinants of information value: synthesizing some general results. Manage. Sci. 27(1), 57±64. Howard, W., Fox, G., Turvey, C., 1996. The economic benefits of new information technology. Working Paper 3/96, Policy Branch. Agriculture and Agrifood Canada, Ottawa. Johnson, S., Holt, M., 1977. The value of weather information. In: Katz, R., Murphy, A. (Eds.). Economic Value of Weather and Climate Forcasts. Cambridge University Press, Cambridge.
111
Johnson, N.L., Kotz, S., 1970. Continuous Univariate Distributions. Houghton±Mifflin, New York. Lave, L., 1963. The value of better weather information to the raisin industry. Econometrica 31(1±2), 51±164. Machina, M., 1987. Choice under uncertainty: problems solved and unsolved. J. Economic Perspectives 1(1), 121±154. Morzuch, B., Willis, C., 1982. Value of weather information in cranberry marketing decisions. Bull. Am. Meteorological Soc. 21, 499±504. Murphy, A., 1994. Assessing the economic value of weather forecasts: an overview of methods, results and issues. Meteorological Applications 1, 69±73. Preckel, P., Loehman, E., Kaylen, M., 1987. The value of public information for microeconomic production decisions. Western J. Agric. Econ. 12(2), 193±197. Robison, L., Barry, P., 1987. The Competitive Firms Response to Risk. MacMillan, New York. Roe, T., Nygaard, D., 1980. Wheat, allocative error and risk: Northern Tunisia. University of Minnesota Economic Development Center Bulletin, University of Minnesota, St. Paul. Schoemaker, P., 1982. The expected utility model: its variants, purposes, evidence and limitations. J. Econ. Lit. 20(2), 529± 563. Stewart, T., Katz, R., Murphy, A., 1984. Value of weather information. Bull. Am. Meteorological Soc. 65(2), 126±137. Tice, T., Clouser, R., 1982. Determination of the value of weather information to individual corn producers. J. Appl. Meteorol. 21(4), 447±452. Vining, K., Pope, C., Dugas, W., 1984. Usefulness of weather information to Texas agricultural producers. Bull. Am. Meteorol. Soc. 65(12), 1316±1319. Warren, H., Leduc, S., 1982. An empirical valuation of meteorological information. J. Appl. Meteorol. 21, 1455±1465. Wilks, D., 1992. Tactical decision making using short-range forecasts: timing alfalfa harvests. Atmospheric Environment Services: Fifth International Meeting on Statistical Climatology.