11.1 UNCERTAINTY
11.2 RISK
11.3 SOURCES OF FARMSYSTEM RISK
11.4 MANIFESTATION OF FARMSYSTEM RISK
11.5 IMPACT OF FARMSYSTEM RISK
11.6 FARMERS' RISKMANAGEMENT STRATEGIES
11.7 FORMAL APPROACHES TO RISKY FARM DECISIONS
11.8 SENSITIVITY ANALYSIS
11.9 STOCHASTIC BUDGETING
11.10 SUBJECTIVE EXPECTED UTILITY
11.11 CERTAINTY EQUIVALENCE
11.12 DECISION TREES
11.13 STOCHASTIC DOMINANCE
11.14 RISKORIENTED MATHEMATICAL PROGRAMMING
11.15 MONTE CARLO SIMULATION
11.16 DIFFICULTIES IN LONGTERM FARM DECISION MAKING
11.17 REFERENCES
'Life is risky. We cannot remember the future.'
Anderson and Dillon (1992)
Uncertainty and risk go hand in hand with farming. They are a pervasive feature of the farm environment. How to handle the risks often associated with uncertainty is the most difficult aspect of farmsystem planning and management. It is the subject of this final chapter. Risk and uncertainty were ignored in preceding chapters. Input prices, output prices, inputoutput coefficients and yields were all assumed to be known with certainty. This is clearly a very strong assumption, particularly relative to longterm planning. For shortterm planning and management, however, the assumption of certainty may often be a reasonable approach. In general, the shorter the planning period, the less the uncertainty. Likewise, the planning and management of a farm subsystem involves less uncertainty than does that of a wholefarm system. Too, much of the uncertainty and risk relating to shortterm planning and management of farm systems is of a repetitive routine nature. Over time, farmers have learnt how to accommodate such routine risk through the use of traditional riskmitigation strategies and a cautionary approach. This is particularly so for resourcepoor small farmers of Type 1 (subsistence) and Type 2 (semisubsistence) whose planning and management might generally be characterized as prudent, circumspect and cautious. Being resource poor and directly dependent on their own production of food staples, Type 1 and (to a lesser extent) Type 2 farmers and their families, more so than others, are at risk of malnutrition, starvation and death if they fail to meet the risks they face from Nature. Caution comes naturally when one's life or family is at stake.
Uncertainty is a pervasive feature of life. It may sometimes exist about the past and this may be relevant to the future. More generally and more importantly, it exists about the future which, unlike the past, cannot be remembered. By its very nature, the future implies uncertainty as a consequence of the irreversible nature of time. Uncertainty is also a function of time in that the further into the future the concern, the greater the uncertainty. There is less uncertainty about tomorrow than there is about a year hence  and less about a year hence than about ten years hence. Consequently, longterm farmsystem planning is far more hazardous than seasonal or annual planning.
Uncertainty is defined as imperfect knowledge. In the context of farm management, it is relevant to managerial decision making about the planning and running of the farm system. Decisions which, because of the influence of forces beyond the farm manager's control, do not have a single sure outcome are known as uncertain decisions. Such decisions have an array of possible outcomes which, it is argued, can be specified/portrayed/viewed by the decision maker in the form of a subjective probability distribution corresponding directly to his or her personal degrees of belief in the occurrence of the possible outcomes. So, if the farmer believes there are n possible outcomes, denoted O_{1}, O_{2},... O_{n} to a particular decision, he or she can attach a subjective probability of occurrence P(O_{i}) to each outcome for i = 1, 2,... n. To be logical, these probabilities must follow the usual rules of probability, viz., the set of outcomes O_{1}, O_{2},... O_{n} should be mutually exclusive and encompass all possible outcomes of the decision so that (1) P(O_{i}) is in the range from zero to one, i.e., 0£ P(O_{i})£ 1; (2) the probability of the ith or jth outcome occurring should equal the sum of their individual probabilities of occurrence, i.e., P(O_{i} or O_{j}) = P(O_{i})+ P(O_{j}), and (3) the probability that one of the outcomes will occur is unity, i.e., P(O_{1} or O_{2}...or O_{n}) = P(O_{1})+ P(O_{2})+...+P(O_{n}) = S P(O_{i}) = 1
These same rules, cast in continuous form, also apply when, unlike the discrete outcome situation given above, the set of possible outcomes constitutes a continuous set. For example, in considering crop yield outcomes in relation to nitrogen fertilizer application, the farmer might consider possible outcomes either in terms of a number of discrete yield intervals or in continuous form as implied by the continuous response function Y = f(N, R) where Y denotes yield per ha, N is nitrogen fertilizer applied per ha and R is rainfall, the latter being a variable beyond the farmer's control which causes yield to be uncertain. The situation is exemplified in Figure 11.1. In the top diagram, response curves are shown for yield response to nitrogen when rainfall is at its best (upper curve) or at its worst (lower curve), i.e., response curves depicting Y = f(N/R) with R, respectively, at its best and worst levels. The vertical distance between the two response curves indicates the range over which the farmer believes yield may vary for any given level of N depending on the amount of rainfall received by the crop. Thus for N = 50, Y may range between 1.5 (if R is at its worst or most ineffective level) and 4.5 (if R is at its best or most effective level). The farmer is assumed (reasonably) to have degrees of belief about the occurrence of these various possible outcomes. These personal degrees of belief for Y values when N = 50 are expressed as subjective probabilities in the bottom half of Figure 11.1  (a) in continuous form on the left and (b) in discrete form (for yield intervals of 0.5t) on the right. The elicitation of such subjective probabilities corresponding to farmers' individual personal degrees of belief is outlined in Section 11.10.3 below.
Relative to Figure 11.1, note that there would be a subjective probability distribution specifiable by the farmer for any relevant level of N. These distributions would, of course, vary in range (and therefore also in location and shape) for different levels of N. The diagram only illustrates the farmer's subjective probability distribution for N = 50. If he or she were to choose to apply 50kg/ha of N, his or her personal degrees of belief in the various possible outcomes would be as depicted in the (continuous or discrete) probability distribution. If the farmer, therefore, were choosing between alternative applications of 25, 50, 75 or 100 kg/ha of N, this choice would devolve or be equivalent to a choice between the subjective probability distributions for outcomes corresponding to these various levels of N. It is in this fashion that decision making under uncertainty corresponds to a choice between subjective probability distributions. Ipso facto, the best choice a decision maker can make under uncertainty corresponds to that choice whose subjective distribution of outcomes most appeals or is most attractive to him or her.
It must be emphasized that the subjective probability distributions of outcomes referred to above are personal to the decision maker. While the shape and location of these distributions may be influenced by others, such as advisers or wizards, or by empirical information from the past or from other locations, these subjective probabilities belong to the decision maker. Because it is his or her decision and consequent responsibility, the degrees of belief pertinent to decision making must also be the decision maker's, not someone else's. It follows that different decision makers may legitimately have different degrees of belief and therefore different subjective probability distributions for the same decision problem. Right or wrong are not words that can be used to describe subjective probability distributions.
FIGURE 11.1  Uncertainty in Cropyield Response to Fertilizer and its Depiction (a) by a Continuous and (b) by a Discrete Subjective Probability Distribution
While uncertainty is always present, risk may not be. Risk is only present when the uncertain outcomes of a decision are regarded by the decision maker as significant or worth worrying about, i.e., when they affect his or her wellbeing (Fleisher 1990, Ch. 2; Robison and Barry 1987, pp. 1314). For a decision problem under uncertainty whose outcomes are regarded as significant, i.e., for a risky decision problem, the risk that is present is specified by the entirety of the decision maker's set of subjective probability distributions for the choicecontingent outcomes that may occur. Only the complete probability distribution (or set of distributions) for the possible (annual) outcomes of a particular (longterm) choice can fully depict the risk which that particular choice entails for the decision maker. Other measures of risk such as the range or mean and variance or coefficient of variation of the distribution of outcomes are (except for special cases such as that of the mean and variance for the Normal distribution or the range for the rectangular distribution) only partial measures of risk.
Depiction of risk by the entire probability distribution of outcomes is somewhat different from everyday usage of the term 'risk' as referring to possible 'bad', 'not so good' or 'negative' outcomes, i.e., outcomes conventionally located on the leftside tail of the probability distribution. Such outcomes are better referred to as downside risk. Conversely, upside risk refers to possible outcomes conventionally located on the rightside tail of the distribution, i.e., to 'good', 'not so bad' or 'positive' outcomes which the decision maker runs the 'risk' of not obtaining.
The risks faced by farmsystem managers may be categorized as coming from two sources: (1) the external environment as it impinges on the farm system and (2) the internal operational environment of the farm system.
Major external sources of risk relate to uncertain turbulence in the natural, economic, social, policy and political environments in which the farm system has to operate (Beal 1996; Fleisher 1990, Ch. 3; Hardaker, Huirne and Anderson 1997, Ch. 1). Of most general relevance and importance are risks associated with the natural environment. Because of its timedependent biological nature, agricultural production is directly dependent on Nature with all its uncertainties and forwardness including, on the downside, the possible vicissitudes of shortterm weather (droughts, floods, frosts, storms, cyclones etc.) and longterm climate (such as climatic change including greater variability due to changes in the Greenhouse Effect), not to mention such natural hazards as earthquakes, tsunamis, volcanic eruptions, landslides, wildfires and the everchanging incidence of pests and disease. All these effects from Nature impinge on yields and, through their effect on market supply, may impinge on prices, both locally and globally.
Risks associated with the farm system's economic environment relate to uncertainty about (1) market (demand and supply) conditions and hence to prices for both inputs and outputs, (2) inflation and interest rates (relevant to longterm planning), and (3) productivity through the availability and merit of new technology. New technology is an interesting source of risk in that, as it is new, farmers will lack experience of it and, being cautious, are likely to subjectively assess it as more risky and less profitable than it possibly is.
The social environment is not, in general, a major source of farmsystem risk. Over time, however, change in education and lifestyles can impact on the availability and competence of farm labour supply. More important is the possible risk of social upheaval and  in the extreme  war which may devastate a country's farm systems, destroy the farm household's support network and  at worst  lead to famine.
Even within a stable sociopolitical environment, changes in the governmentdecreed policy environment may be a significant source of risk to farmers. Policy or institutional areas of particular relevance are those relating to: commodity prices and marketing; the availability and cost of credit, water rights and other inputs; the availability of public infrastructure; environmental standards; safety and health standards; labour laws; land tenure; export and import regulations; exchange rate controls; and general government reactions to ongoing market globalization with increasing competition and pressures for deregulation and economic restructuring. There are two dimensions to such policyrisk possibilities: first, uncertainty about what changes may be legislated and, second, uncertainty about the extent to which legislated changes will be enforced. Policy risk may impact on the farm household's income either through effects on yields, on total output level, or on input or output prices.
The final broad external source of farm risk comes from uncertainty about the political environment, i.e., from any marked change in the political ideology holding sway such as, e.g., moves from a socialistic centrally planned to a capitalistic freemarket system or vice versa. Such changes are likely to be dramatic in their effects on the type and scale of farming systems in use. They are particularly pertinent to longrun farm planning.
All the above external sources of risk impinge more or less equally on those farm systems which share the same external environment. In contrast, sources of risk internal to the farmhousehold system impinge, in the main, only on the particular farm. The major internal sources of risk relate: (1) to the health of the farm household's members; (2) to the interpersonal relations between farmhousehold members as influenced by personality, changing values, attitudes and aspirations; and (3) to the approach followed by the farm manager relative to (a) the conservation or degradation of farm resources (leading to resource and ecological risk), (b) the use of credit to finance the farm's operation and development (leading to financial risk), and (c) the intergenerational transfer of the farm (leading to succession risk). While the first two of these internal sources of risk (i.e., health and family relationships) relate particularly to the short term, the third set of sources is of more longterm relevance.
Beyond the above sources of risk that impinge directly on the farm system itself, it should be noted that the farm management analyst or adviser, regardless of the Field or Mode of operation (Sections 2.1.7 and 8, respectively), faces an additional source of risk. This is that, despite the best of efforts, the data and/or models used in analysis and diagnosis may be inadequate or incorrect. He or she should therefore always exercise caution in the interpretation of analysis and the provision of advice. Too, the analyst should recognize that the risks faced by farmers from the above external and internal sources are not only pervasive but also additive. This means that, in considering a new farm activity or technique, the important question may be how it adds to (or subtracts from) the prevalent risk (Hardaker, Huirne and Anderson 1997, Ch. 11).
Whatever its source, farmsystem risk manifests itself as a set of possible outcomes across each of the alternative choices the farmer may take in managing the farm system or its subsystems. So long as the outcomes for each alternative are defined so as to be mutually exclusive and exhaustive, their chance of occurrence can be specified by the farmer in the form of a subjective probability distribution. By definition, these risky outcomes are of economic significance, i.e., they are significant net gains or losses measured, as appropriate, either in financial or other terms. They may relate to the wholefarm system of Order Level 10 or to lowerlevel subsystems, particularly crop and livestock enterprises and activities of Order Levels 4 to 7 (Figure 1.2).
The critical economic variables through which uncertainty in the farm's external or internal environment is translated into farmsystem risk are those which determine the net economic outcome of a decision. For commercial farmers, these variables are product prices, unit yields and aggregate yields, the prices and quantities used of variable inputs, and fixed costs. For purely subsistence farmers with no market contact, input or output price risk is irrelevant and risk to their subsistence livelihood is manifested only through uncertainty in unit and aggregate yields. So, while commercial farmers may suffer risk from all the sources discussed in Section 11.3 above, risk for subsistence farmers is largely manifested through yield uncertainty arising from the vagaries of Nature and, if they are illserved politically, from the social upheaval of civil unrest and war. Today, however, purely subsistence farms are rare. Nearly all small farms have some market contact. Small farms of Type 1 (subsistence) and Type 2 (semisubsistence) are therefore generally mainly affected by Natureinduced yield risk but, to varying degree, must also face risk arising externally from the economic, social, policy and political environments, as well as internal risks related to the health and relationships of farmhousehold members.
Risk in shortterm decisions. Relative to shortrun decisions, it is generally product prices and yields which are uncertain and thus most important in determining risk. This is because, in the shortrun context of seasonal or annual decisions, the quantity of variable inputs used is usually under the farmer's control and the price of variable inputs and the level of fixed costs are usually known to the farmer at the time of decision, i.e., at the start of the production process; it is product yield and price at the termination of the production process at harvest that are uncertain. The fertilizerchoice situation depicted in Figure 11.1 provides an example.
Risk in longterm decisions. In the context of longterm decisions, all yield, price and cost variables are likely to be uncertain. As illustrated in Section 10.12, yield profiles over the life of the crop form the basis of investment planning for longterm crops such as tea, cacao, coffee etc. Such profiles, as determined by agronomic factors, can be predicted with a high degree of confidence in the absence of uncertain events. However, in any particular year, actual achieved yield may depart significantly from expected yield due to the effect of such unpredictable events as cyclones, storms, tidal waves, droughts, frosts, outbreaks of disease, attacks by monkeys or other pests. Compared with longterm yield patterns, commodity prices of longterm crops are generally more variable and uncertain. Empirically, longterm prices can behave in several ways. Some commodities have no apparent trend over time and are completely unpredictable. However, most commodity prices have either a downward or an upward secular trend (as well as narrow or broad variations from this trend)  a downward trend in the case of certain old commodities which are gradually being replaced by substitutes (e.g., cinchona, sisal), or an upward trend in the case of 'luxury' commodities or those which can be produced in only a few suitable locations (e.g., cardamom, cloves). Regardless of their trend, some commodity prices are subject to a regular and predictable cycle superimposed on such trend (e.g., coffee, cacao, pepper) but the variations from this trend are usually not predictable. Due to inflation, longterm costs normally tend to increase over time in nominal terms. But superimposed on this fairly predictable cost stream, abnormal costs can also arise. Without notice, the activity might be devastated by a cyclone or drought and have to be reestablished; or routine normal input costs might temporarily rise to high levels because of shortage due to, e.g., nonarrival of the supply ship.
The obvious impact of farmsystem risk is to make management of the farm more difficult. The outcome of a management decision under risk cannot be sure. Ex ante, outcomes may be good or bad. Ex post, with hindsight, there are likely to be regrets about what might have been. All that the farmer can do, given the imperfect information available, is to make that choice among the available risky alternatives which most appeals to him or her given his or her preference for outcomes and degrees of belief in their occurrence. Such a normative approach is based on subjective expected utility theory as outlined in Section 11.10 below. For the farm manager, it implies the difficult task of specifying subjective probability distributions of outcome possibilities relative to each choice alternative and then choosing between these distributions.
A second impact of risk, especially for small farmers of Types 1 and 2, is that it generally makes them cautious in their decision making. Farmhousehold survival demands that greater cognizance be taken of possible adverse outcomes (i.e., downside risks) than of possible good outcomes. While good outcomes (i.e., upside risks) are inherently attractive, for resourcepoor small farmers this attraction must be expected to be more than outweighed by concern for the possibly disastrous impact of adverse outcomes. Downside risks challenge farmsystem survival, particularly if a series of adverse outcomes should occur. In the sense of Darwinian evolution with its concepts of natural selection and adaptation to the environment over time, farmsystem survival is enhanced by a prudent, circumspect and cautious approach to risky decisions on the part of the farmer. It is not surprising, therefore, that farmers (in common with most other decision makers) generally exhibit risk aversion (rather than risk neutrality or risk preference) in their decision making. Given the opportunity, as outlined in Section 11.10.1, they would be prepared to pay some risk premium in order to avoid risk and achieve some sure or guaranteed outcome.
Finally, in terms of impact, there is no doubt that the agricultural sector and small farms in particular are generally more susceptible to downside risk than other sectors of the economy. There are two reasons for this. First, while just as subject as other sectors to risk from the economic, social, policy and political environments, agriculture is, because of its timedependent biological basis, far more exposed to downside risks from a potentially hostile and uncontrollable Nature. Second, farmers (in particular, resourcepoor small farmers) generally do not have easy access to such formal institutional avenues of risk mitigation as purchased insurance and futures markets. This is because of agriculture's dispersed nature and largely atomistic smallscale production structure based on Type 1 (subsistence) and Type 2 (pancommercial) small farms producing mainly food staples for home consumption and having limited market orientation.
As already noted, farmers are cautious and circumspect in the face of downside risk. They exhibit risk aversion, not neutrality or preference (Section 11.10.1 below). This is particularly so for resourcepoor small farmers whose livelihood can often be at stake from risk. Their basic riskmanagement strategy of caution is manifested in a wide variety of operational strategies aimed at risk mitigation (Anderson and Dillon 1992, Sections 2.2.1 and 3.3). As summarized in Table 11.1, for Type 1 (subsistence) and Type 2 (partcommercial) small farmers, these strategies are most evident in relation to production or yield risk  a not unexpected feature given the farmers' close dependence on Nature. Fewer strategies are directly oriented to meeting price, marketing and financial risks. Unlike larger commercial family farms and estates (i.e., farms of Type 5 and Type 6, respectively), Type 1 and Type 2 farmers do not normally have available to them more formal marketbased institutional approaches to risk management such as a bank line of credit or overdraft facility, crop or livestock insurance, forward pricing through price, futures or options contracts (Anderson and Dillon 1992, Sections 2.2.2 and 3.4; Calkins and DiPietre 1983, Ch. 9; Fleisher 1990, Ch. 7) or market guarantees through vertical integration.
TABLE 11.1  An Overview of Small Farmers' Riskmanagement Strategies
Functional orientation 
Strategy 
Type of risk reduced 

Production

Use of stable enterprises 
Yield, technology, policy


Use of tolerant cultivars 

Diversification 


in crops 


in livestock 


within seasons 


across seasons 


across space 

Maintain flexibility 


over time 


in durable assets 

Keep reserves seed 


seed 


fodder 

Use riskreducing inputs 

Shareleasing 

Assess new technology 

Seek information 

Marketing

Spread sales over time 
Price


Arrange alternative outlets 

Seek barter opportunities 

Financial

Maintain high equity ratio 
Financial, yield, price


Maintain credit worthiness 

Maintain a cash reserve 

Maintain fungible assets 

Maintain social network 

Offfarm employment 
Not all small farmers, of course, use all the riskmanagement strategies listed in Table 11.1. Their use varies by agroecological region. But whatever combination of risk strategies is used, they form an integral part of the farming systems developed by small farmers over the centuries in particular environments. Particularly in relation to production shocks, riskmanagement strategies used by small farmers differ according to when they are applied (Matlon 1991). Major emphasis is on ex ante approaches designed to insure the farm household against downside shocks before they occur. Once a shock is apparent, interactive approaches involving the reallocation of resources are used to minimize adverse impact. And after a shock has occurred, ex post approaches of a compensatory nature are applied to ensure farm household survival. Table 11.2 illustrates, in a levelbytime framework, the various approaches to risk management typically taken by small farmers in the semiarid tropics.
TABLE 11.2  Example of Classification by Level and Time of Riskmanagement Strategies used by Small Farmers in the Semiarid Tropics
Level 
Time frame 

Ex ante 
Interactive 
Ex post 

Plant 
Varietal selection for stress resistance or tolerance 
Replanting with earlier maturing varieties 

Animal 
Use of tolerant breeds and types 
Sell to reduce stocking rate 

Plot

Early/staggered planting dates 
Change crops with replanting 
Grazing of failed plots 
Low hill density 
Change plant density through thinning or replanting 
Late planting for forage production 

High seeding rate 



Intercropping 



Runoff management 



Delayed fertilizer application 



Farm

Diversified cropping 
Shirting crops between land types 
Local nonfarm work 
Landtype diversification 



Plot fragmentation 



Household, village, region

Cereal stocks 
Offfarm work 
Cereal rationing 
Livestock as fungible assets 

Livestock sales Migration work 

Social networks 

Food transfers by gift or as aid 

Nonfarm work 


Source: Matlon (1991).
In the droughtprone semiarid tropics, diversification in crops, in varieties and in their toposequence location plays a large part in ex ante risk management. Crop diversification involves cropping different crops across dispersed plot areas as well as mixing crops in the same plots. Such intercropping improves yield and income stability to the extent that crop mixtures yield better than monocrops under stress conditions, reduce the incidence and buildup of pests and diseases, and exhibit compensatory yield behaviour due to differences in crop physiology, phenology or architecture. Varietal diversification in terms of days to maturity and variability in resistance or tolerance to biotic stresses reduces the chance of total crop failure due to periodspecific stresses. Likewise, by diversifying plot locations across the soil toposequence, farmers are able to reduce plotyield covariation and thus reduce aggregate production variability.
Interactive approaches to semiarid risk management, as listed in Table 11.2, rely on sequential decision making as downside shocks unfold. At the start of each cropping season, farmers have subjective degrees of belief about the onset, amount and duration of rains and the likelihood of pest and disease occurrence. As the season unfolds, farmers revise these degrees of belief and sequentially adjust their cropping pattern and cultivation practices as necessary.
Ex post compensatory strategies for risk management used by farmers usually relate more to the farm household qua household than to farm production activities. Nonfarm work, sale of fungible assets such as cattle, rationing of cereal consumption and gifts of food from other households or relief agencies are common approaches used by small farmers to maintain income and sustain household consumption following crop failure.
Being rational, farmers typically make their risky decisions in a reasoned way. This, it may be argued, implies that, on the basis of their experience, traditional knowledge and whatever other information is available to them, farmers specify: (1) the alternative choices open to them; (2) the set of uncertain outcomes associated with each of these alternative choices; and (3) their personal subjective probability distribution for each of these sets of outcomes (reflecting for each farmer, his or her degrees of belief in the possible occurrence of these uncertain outcomes). Personal judgement is then exercised by the farmer to choose that alternative which, for him or her, has the most preferred/attractive probability distribution of outcomes. Such a process of choice is generally carried out by the farmer in an implicit or informal rather than an explicit or formal manner. This is particularly so for Type 1 (subsistence) and Type 2 (partcommercial) small farmers. It is a little less so for commercially oriented farmers of Types 3, 4 and 5 (independent specialized, dependent specialized and large family farms, respectively) and of Type 6 (estates). These may sometimes use more explicit formal procedures either themselves or through the services of professional advisers. In particular, this is the case relative to longterm risky decisions in estate management.
As the above reference to professional farm management advisers implies, when such specialists operate professionally in Mode 4 (i.e., in prescriptive mode, Section 2.1.8) in Field A (i.e., in advising farmers on a participatory basis, Section 2.1.7), they have at their disposal a variety of formal techniques to assist the farmer's evaluation of risky decisions and guide his or her risky choices (Dillon and Hardaker 1993, Ch. 8; Hardaker, Huirne and Anderson 1997). Most relevant of these formal approaches to risky choice are sensitivity analysis, stochastic budgeting, subjective expected utility analysis (including certainty equivalent analysis and decision trees, and stochastic dominance analysis), riskoriented mathematical programming and Monte Carlo simulation. These all somewhat overlap and interrelate. Except for sensitivity analysis, they can all be cast in an expected utility framework based on the decision maker's personal preferences among outcomes and his or her personal degrees of belief in their occurrence. Nonetheless, each is outlined separately in turn below. The presentation is expository and nonalgebraic. More comprehensive discussion of risky decision making in agriculture is provided, in order of increasing analytical orientation, by Hardaker, Huirne and Anderson (1997), Anderson, Dillon and Hardaker (1977), Rae (1994, Chs 9, 10 and 11), Dillon and Anderson (1990, Ch. 7) and Robison and Barry (1987).
Conditional or parametric budgeting (Section 4.7) forms the basis of sensitivity analysis which aims to assess how sensitive a decision's outcome may be to changes in those variables affecting the decision (Belli 1996, Ch. 10). Sensitivity is quantified by testing the effect of variations in selected cost and benefit variables on budgeted outcomes of the decision. These variations may be across the particular range of values felt to be relevant for each variable or may be some arbitrary percentage change above and below the value used in the base (nonparametric) budget. This base budget will generally be based on either the expected (i.e., mean) or most likely (i.e., modal) values of the variables entering the budget.
Tables 11.3, 4 and 5 provide an example of sensitivity analysis applied to assessing the financial viability of smallscale sponge farming in Micronesia (Adams, Stevely and Sweat 1995). Table 11.3 shows the base budget for a 0.4 ha farm with a threeyear harvest rotation, i.e., 0.4/3 = 0.133 ha are harvested each year. Since in this particular case the base budget is meant to represent the likely annual performance of the enterprise once established, the budget entries are most likely or modal values as specified or agreed by the farmer. The budget indicates that with a survival rate of 90 per cent from planting to harvest and a marketability of 80 per cent, the 0.4 ha farm would give a pretax net annual return to the owner for his or her management, capital and risk of $784. The budget also implies that, other things unchanged, the minimal survival rate from planting to harvest that would allow total production costs, excluding opportunity costs, to be recovered is 42.8 per cent. This is the breakeven survival rate (in budget terms) or, to use the jargon of sensitivity analysis, the switching value for this variable if the initial assumption is of a 90 per cent survival rate.
Table 11.4 shows the sensitivity of budgeted net return, production cost per sponge sold and breakeven survival rate to changes in four key variables. These four changevariables were chosen on a priori grounds for their expected relevance to the budget calculations. Likewise the range of values over which they are varied reflects subjective judgement of their possible range of variation. The budgeted results of the sensitivity analysis show, e.g., that annual net return to the owner's management, capital and risk may vary from $416 to $4 984 in terms of possible marketprice variation, from $224 to $960 relative to survival rate, from $484 to $1 084 relative to marketability percentage and from $676 to $1 166 relative to total variable cost. The results also indicate that the switching value of survival rate between financial viability and nonviability is more sensitive to market price than to the other variables.
The sensitivity analysis of Table 11.4 is based on oneatatime change in the variables. In the real world, however, all the variables would be subject to simultaneous change from their expected values. This more realistic situation is appraised in Table 11.5 in terms of the budgeted outcomes if each of the variables were to simultaneously occur at their most favourable, their most likely and their most unfavourable levels. This indicates that if the variables were to occur at their best possible level, the owner's return to his or her management, capital and risk would be $6 413; at the other extreme, if the variables took their worst possible level, the return would be a loss of $924.
TABLE 11.3  Annual Enterprise Budget for a 0.4 Ha Sponge Farm with 0.133 Ha harvested Annually
Revenue^{a} 
$2400  
Costs of production 
 

Variable costs: 
 


Oil and gas 
200 


Boat and motor maintenance 
182 


Marketing costs 
60 


Supplies 
98 

Total variable costs 
540  

Fixed costs: 
 


Interest^{b} 
0 


Depreciation 
116 

Total fixed costs 
116  

Total production costs 
656  
Net return before tax to owner for management, capital, labour and risk 
1 744  
Opportunity cost of owner's labour 
960  
Net return before tax to owner for management, capital and risk 
784  
Cost per sponge sold 
0.273  
Breakeven survival rate (%) 
42.8 
^{a} Based on a survival rate of 90 per cent and a marketability of harvested sponges of 80 per cent, yield of 2 400 units and a market price of $ 1.00 per unit.^{b} Farmer equity of 100 per cent is assumed in the necessary capital investment of $650.
Source: Adams, Stevely and Sweat (1995).
It should be noted that only one alternative  investment in a 0.4 ha sponge farm  is explicitly considered in the above example. The implied decision is between this investment and doing nothing, i.e., maintaining the status quo of having, say, the money invested at a known rate of interest.
The above example well illustrates the pros and cons of sensitivity analysis. On the positive side, (1) in a budget context, it is readily carried out using spreadsheet analysis on a personal computer; (2) it gives an indication of the range of possible outcomes; (3) it indicates the relative sensitivity of outcomes to the changevariables assessed; (4) it enables the determination of switching values for key changevariables; and (5) it may pinpoint the need for additional information on some variables. In general, however, these positive aspects are outweighed by the three major limitations of sensitivity analysis: (1) it does not take into account the probability of occurrence of changes in the variables; (2) nor does it take into account correlation among possible changes in the variables; and (3) if, as often commonly used, the values of the changevariables are varied by standard (arbitrary) percentages, e.g. ±10 or ±20 per cent, these changes may bear no relation to the likely variability of the changevariables. The lack of account for the probability and correlation of change in the variables is particularly damaging. Thus, e.g., Table 11.4 gives net return outcomes depending on market price but no indication of their probability of occurrence which, of course, parallels the (unspecified) probability distribution for market price. Likewise, the outcomes presented in Table 11.5 for simultaneous variation in the four changevariables should also be conditioned by their joint probability of occurrence which parallels the joint probability of the four changevariables. The usual practice of varying only one variable at a time, e.g., as in Table 11.4, is justified only if the changevariables are uncorrelated; otherwise the changevariables should be considered jointly as in Table 11.5. In the context of the above example, survival rate and marketability percentage are probably negatively correlated with market price. As a result, the sensitivities shown in Table 11.4 may be exaggerated since they do not allow for counterbalancing variation in these three variables.
TABLE 11.4  Summary of Sensitivity of Annual Net Return^{a}, Cost per Sponge, and Breakeven Survival Rate to Changes in Four Key Variables^{b}
Variable 
Net return 
Production cost per sponge sold 
Breakeven survival rate 
Market price per unit ($) 



0.50 
416 
27.3 
60.7 
0.75 
184 
27.3 
48.8 
1.00 
784 
27.3 
42.8 
1.25 
1 384 
27.3 
39.3 
1.50 
1 984 
27.3 
36.9 
1.75 
2 584 
27.3 
35.2 
2.00 
3 182 
27.3 
33.9 
2.25 
3 784 
27.3 
32.9 
2.50 
4 384 
27.3 
32.1 
2.75 
4 984 
27.3 
31.5 
Survival rate (%) 



75 
224 
35.7 
42.8 
80 
408 
32.7 
42.8 
85 
584 
29.7 
42.8 
90 
784 
27.3 
42.8 
95 
960 
25.5 
42.8 
Marketability (%) 



70 
484 
31.2 
45.4 
75 
634 
29.2 
44.0 
80 
784 
27.3 
42.8 
85 
934 
25.7 
41.8 
90 
1 084 
24.7 
40.8 
Total variable cost (% change) 



20 
1 166 
21.4 
32.4 
10 
838 
25.0 
41.4 
0 
784 
27.3 
42.8 
+10 
730 
30.0 
44.3 
+20 
676 
32.0 
45.8 
^{a} Net return to the owner's management, capital and risk.
^{b} Values in bold represent results for the baseline set of assumptions.Source: Adams, Stevely and Sweat (1995).
TABLE 11.5  Effect on Annual Net Return of Simultaneous Variation in the Four Key Variables of Table 11.4
Level of variable^{a} 
Variable 
Net return^{b} 

Survival rate 
Marketability 
Unit price 
Total variable cost 

(%) 
(%) 
($) 
($) 
($) 

At worst 
75 
70 
0.50 
648 
924 
Most likely 
90 
80 
1.00 
540 
784 
At best 
95 
90 
2.75 
432 
6413 
^{a} The worst and best extreme values from Table 11.4.
^{b} Net return to the owner's management, capital and risk.Source: Adams, Stevely and Sweat (1995).
As its name implies, stochastic or risk budgeting is carried out by attaching probabilities of occurrence to the possible values of the key variables in a budget, thereby generating the probability distribution of possible budget outcomes (Dillon and Hardaker 1993, pp. 169172; Hardaker, Huirne and Anderson 1997, pp. 120126). The probabilities used should be those of the decision maker, i.e., his or her personal or subjective probabilities based on experience, intuition and/or any other available information. As an example, consider the budgets relative to survival rate presented in Table 11.4. In this situation, the only variable assumed to be risky is survival rate. Accordingly, if the farmer's personal probabilities for survival rate, denoted by P(s_{i}), are as shown in Table 11.6, these also constitute the probability distribution for budgeted annual net return to the farmer's management, capital and risk. This distribution has a modal (i.e., most likely) value of $784 and, as calculated in Table 11.6, an expected or mean value of $639.
TABLE 11.6  Stochastic Budget relative to Survival Rate for the Spongefarm Investment of Table 11.4
Survival rate 
Probability of survival rate 
Net return^{a} 
Cumulative probability of net return  
Central value 
Interval 
(P(s_{i})) 
(R) 
(P(R £ R*)) 
(%) 
(%) 

($) 

75 
72.5 < s £ 77.5 
0.10 
224 
0.10 
80 
77.5 < s £ 82.5 
0.15 
408 
0.25 
85 
82.5 < s £ 87.5 
0.25 
584 
0.50 
90 
87.5 < s £ 92.5 
0.40 
784 
0.90 
95 
92.5 < s £ 97.5 
0.10 
960 
1.00 
Expected net return = S P(R_{i})R_{i} = (0.10)224 + (0.15)408 + (0.25)584 + (0.40)784 + (0.10)960 = $639.20 
^{a} Net return to the owner's management, capital and risk.
As shown respectively in Table 11.6 and Figure 11.2, the distribution of net return can also be listed or graphed in the form of a cumulative probability distribution or cumulative distribution function (CDF). Denoting net return by R, this cumulative distribution shows the probability that R will be less than or equal to any nominated value R*, i.e., P(R<R*) and, ipso facto, the probability that R will exceed any nominated value R*  which probability is necessarily equal to one minus the probability that R will be less than or equal to R*, i.e., P(R>R*) = 1  P(R £ R*). Thus, reading from the cumulative distribution sketch of Figure 11.2, the probability that annual net return will be no more than $500 (i.e., R* = 500) is P (R £ R* = 500) = 0.364. Conversely, the probability that annual net return will exceed $500 is 1  0.364 = 0.636.
Just as with sensitivity analysis, stochastic budgeting with discrete probabilities is readily conducted via spreadsheet analysis on a personal computer. More generally, specialist software addins such as @Risk (Palisade Corporation 1997; Winston 1996) for use with spreadsheet software such as Excel or Lotus 123 can be used. With @Risk, a wide variety of standard probability distributions such as the Normal, Beta, Gamma, triangular or rectangular may be used, as well as specific distributions specified by the user. An example of the use of @Risk for stochastic budgeting is provided by Hardaker, Huirne and Anderson (1997, pp. 121126).
FIGURE 11.2  Cumulative Probability Distribution of Annual Net Return (R) for the Stochastic Budget of Table 11.6
While the simple example of stochastic budgeting presented in Table 11.6 above is based on enterprise budgeting, stochastic budgeting can be applied to any type of budget whether it be of a wholefarm or partial, cashflow or profit, single or multiperiod nature.
Finally, it should be noted that stochastic budgeting is in fact a simple form of Monte Carlo simulation analysis (Section 11.15), the only difference being that the latter often involves more elaborate modelling of the decision alternative being simulated.
11.10.1 Personal preference or utility
11.10.2 Utility function elicitation
11.10.3 Probability elicitation
11.10.4 Example of subjective expected utility analysis
The term utility is used here in the sense of satisfaction. Subjective expected utility theory is a logical approach to risky decision making based on a few eminently reasonable axioms such as, e.g., if a is preferred to b and b is preferred to c, then a will be preferred to c (Anderson, Dillon and Hardaker 1977, Ch. 4). These axioms imply: first, that a personal utility function U (such as the curve ABC in Figure 11.3) can be ascribed to a decision maker such that an amount of utility U(O_{i}) can be associated for him or her with the outcome O_{i}, second, that if O_{i} is preferred to O_{i}, then U(O_{i})>U(O_{j}); third, that if O_{i} is uncertain (i.e., not sure), then the decision maker can specify a subjective probability for its occurrence corresponding to his or her degree of belief in its occurrence so long as the set of outcomes {O_{i}} associated with a particular choice is mutually exclusive and exhaustive; and, fourth, that the utility of the particular choice having the set of risky outcomes {O_{i}} is equal to the expected utility of the set of outcomes, i.e.,
U({O_{i}}) = EU({O_{i} })
where E denotes statistical expectation so that, e.g., for the discrete case where O_{i} (i = 1,2,...n) has probability P(O_{i}):
U({O_{i}}) = EU({O_{i}}) = U(O_{1})P(O_{1}) + U(O_{2})P(O_{2}) +...+ U(O_{i})P(O_{i}) +...+ U(O_{n})P(O_{n}) = _{}
Subjective_{} expected utility theory is thus remarkable in that it brings together and integrates the three crucial elements of risky decision making: first, the decision maker's personal preferences about possible outcomes; second, the decision maker's personal degrees of belief in the occurrence of possible outcomes; and, third, through the use of his or her own personal preferences and probabilities, the decision maker's personal responsibility and accountability for whatever decision is taken.
Figure 11.3 illustrates the concept of a utility function and expected utility for the typical case of a riskaverse decision maker. The utility function U is depicted by the curve ABC which, being concave, has positive but diminishing slope to the right, indicating (1) that more is preferred to less but, because of satiation, with diminishing marginal utility and (2) that the decision maker is risk averse. Reading from the curve, it can be seen, e.g., that an outcome yielding $3 000 has a utility value of 2.3.
While raw outcomes are here measured in money terms, any relevant measure such as, e.g., bags of rice, may be used as appropriate. Too, it is important to note that the utility scale is always arbitrary, analogous to a temperature scale so that any function U* will serve as well as the function U so long as U* = aU + b with a > 0. Thus it makes no sense to say that one outcome, e.g., would give twice as much utility as another. All that can be said is that one outcome exceeds another in utility, i.e., is preferred to another. Nor, because of the personal nature of utility, can comparisons be made between one person's utility values and those of another person. Also, the utility assessment of alternative farm decisions must usually be on an aggregate wholefarm, project or enterprise basis, not on a technical unit basis such as per ha or per tree or per head. Only if each alternative is of the same size in terms of technical units is it appropriate to make utility comparisons on a technical unit basis. Likewise, for utility analysis, financial outcomes should be measured as net returns or profit, not as gross margins since maximizing U(TGR  TVC  TFC) is not the same as maximizing U(TGR  TVC) = U(TGM). Note also that if an alternative involves n technical units, each giving a payoff of $X for a total payoff of $nX, diminishing marginal utility implies that U(nX) < nU(X).
FIGURE 11.3  Illustration of the Concept of a Utility Function and Expected Utility for a Riskaverse Decision Maker
Suppose the decision maker whose utility function is depicted in Figure 11.3 has to choose between two alternatives, one being a sure prospect offering a sure gain of $2 600 and the other a risky prospect offering a 50:50 chance of either $1 000 or $5 000. From the graph of Figure 11.3, the sure gain of $2 600 has a utility value of 2.12. The utility of the risky prospect ($1 000, $5 000; 0.5, 0.5) is given by its expected utility. This is calculated as U ($1 000)(0.5) + U ($5 000)(0.5) which, from Figure 11.3, is equal to (1.2)(0.5) + (2.8)(0.5) = 2. Since 2.12 > 2, the alternative offering a sure gain of $2 600 would be preferred to the risky prospect. This is so even though the expected value of the risky prospect's raw outcomes, i.e., its expected money value (EMV) of ($1 000)(0.5) + ($5 000)(0.5) = $3 000, is greater than the sure prospect's money value of $2 600.
Risk attitude
Figure 11.3 also illustrates the concepts of risk aversion and certainty equivalence (Hardaker, Huirne and Anderson 1997, Ch. 5). The decision maker's risk aversion is shown by the fact that, although the risky prospect has an EMV of $3 000, because of its risky nature its utility value of two is less than that of a sure $3 000 which, reading from the utility curve of Figure 11.4, has a utility value of 2.3. In fact, corresponding to point B on the utility curve, the risky prospect's utility value of two is the same as that of an outcome of $2 400. In other words, for this particular decision maker, the risky prospect ($ 1 000, $5 000; 0.5, 0.5) is equivalent in utility terms to a sure prospect of $2 400. This amount of $2 400 is thus referred to as the certainty equivalent of the risky prospect. In general, subjective expected utility theory implies that, for any risky choice faced by a decision maker, it will always be possible to specify some sure prospect which has the same utility value as the risky prospect. In other words, the decision maker would be indifferent between taking the risky choice and receiving the sure prospect. This sure prospect is known as the certainty equivalent (CE) of the risky choice. Moreover, since different decision makers have different utility functions, they will also legitimately have different CEs for the same risky prospect.
The difference between the mean or expected value (EV) of a risky prospect and its CE (i.e., EMV  CE in the case of a money prospect) is the decision maker's risk premium for the risky prospect. For the example of Figure 11.3, the risk premium is given by the distance BD implying a risk premium of $3 000  $2 400 = $600: this decision maker would be prepared to give up 600 'expected $' in order to avoid the risky prospect. Another decision maker could legitimately have a different risk premium for the same risky prospect.
If the decision maker is risk neutral, his or her utility function is linear, thus implying a risk premium of zero as would be the case, e.g. if the line ADC of Figure 11.3 were the decision maker's utility function. Conversely to the case of risk aversion, risk preference implies (1) a convex utility curve, i.e., one having positive but increasing slope to the right corresponding to increasing marginal utility, and (2) a negative risk premium since with risk preference the decision maker would be prepared to pay a premium to face the risk opportunity.
The above relationships between the shape of a decision maker's utility function, risk attitude, risk premium and marginal utility are summarized in Table 11.7.
TABLE 11.7  Relationships between the Shape of the Utility Function, Risk Attitude, Risk Premium and Marginal Utility
Shape of the utility function U(X) 
Attribute 

Risk attitude 
Risk premium 
Marginal utility 

Concave 
aversion 
positive 
diminishing 
Linear 
neutrality 
zero 
constant 
Convex 
preference 
negative 
increasing 
As noted in Section 11.5, most decision makers, particularly including small farmers, are risk averse. Perhaps the only situation in which small farmers may behave in a risk preferring way is when their household faces a food crisis. In such circumstances they may (be forced to) take decisions (such as selling assets, borrowing funds, depleting resources) which place the household's future livelihood from the farm in jeopardy. Generally, however, risk preference is sufficiently rare that it need not concern the farm management analyst. Too, for many risky farm decisions where the sets of possible outcomes are not vastly different in size or probability, it may often suffice to assume risk neutrality and assess alternative choices in terms of their expected or mean outcomes.
A number of methods is available for the elicitation of a decision maker's utility function. These are all based on his or her responses to a chain of questions involving choice between hypothetical risky prospects. Details of the methods (along with much fuller discussion of the concept of a utility function) are given by Anderson, Dillon and Hardaker (1977, Ch. 4) and Hardaker, Huirne and Anderson (1997, Chs 3, 4 and 5).
In general, being based on mind experiments rather than realworld risky prospects, elicitation is not always reliable. However, this is not a problem relative to small farmers. Except perhaps for case studies involving representative small farms, it would not be justifiable to attempt to elicit a small farmer's utility function. Decision guidance to such farmers on an individual basis will generally be more efficacious if based on the certainty equivalent approach (Section 11.11). Only for larger commercial farms and estates, and then only for significant riskychoice problems, is it likely that direct elicitation of the manager's utility function to provide a basis for risky choice would be warranted. More generally, risk appraisal pertinent to farmers or farming systems in different agroecological zones can be based either (i) on an arbitrary assumption of a general riskaverse utility function such as 'Everyman's Function' U = log_{e}W where W is the risky outcome, or (ii) on methods such as stochastic dominance analysis (Section 11.13) and riskoriented mathematical programming (Section 11.14), the software for which can enable sensitivity analysis to be carried out on a personal computer relative to varying degrees of risk aversion and forms of the utility function. A particular case for which such approaches are relevant is in the evaluation of trial data for new technology, e.g., for the evaluation of new crop varieties before their release to farmers.
Just as for the elicitation of small farmers' utility functions, the elicitation of small farmers' subjective probability distributions for relevant risky events is generally not a practical possibility. There are too many small farmers, each facing too many risky decisions  for each of which (unlike for his or her utility function which remains the same for different decisions) each farmer will have different sets of degrees of belief. Only for case studies of representative situations might small farmers' probabilities be elicited. Conversely, probability elicitation could be a normal element of specialist services used by the managers of larger commercial farms and estates in their appraisal of important risky choices.
A farmer's degrees of belief in uncertain events are usually best elicited in the form of discrete probability distributions using a visual display procedure. As illustrated by Figure 11.4, the procedure is based on the decision maker allocating counters across the various possible eventsituations in accord with his or her degrees of belief in their occurrence. The elicited relative frequency of the counters across the various events then corresponds to the decision maker's subjective probability distribution for the events considered.
Based on the spongefarm example of Section 11.8, Figure 11.4 shows the farmer's elicited frequency distribution for the jointevent possibilities of survival rate percentage (s_{i}) and marketability percentage (m_{j}). Dividing each of these frequencies by the total number of counters used (i.e., by 100 in this case) gives the farmer's subjective probability for each joint event (s_{i} and m_{j}). Thus the joint event of a 90 per cent survival rate and marketability of 80 per cent has a joint probability of 15/100 = 0.15. Likewise, the row and column frequency totals give the farmer's marginal probability distribution, respectively, for survival rate and marketability, i.e., for each of these considered without regard to the other. Thus the probability of marketability being 70 per cent is 12/100 = 0.12. As a practical matter, in allocating counters to complete a display such as depicted by Figure 11.4, it will usually be easier if the marginal frequencies are allocated first. Of course, if only a single variable is involved, only a oneway rather than a twoway display needs to be completed. If more than two variables need to be considered jointly, a multistage display procedure needs to be used. As an example, Figure 11.5 shows the second and finalstage display for the elicitation of the farmer's joint probabilities P(s_{i}, m_{j} and p_{k}) for survival rate, marketability and market price (p_{k}) in the spongefarm example. The firststage display is that of Figure 11.4. In the secondstage display of Figure 11.5, the 5 x 5 = 25 joint events of Figure 11.4 are listed out to give all their possibilities of joint occurrence with five levels of market price, i.e., a total of 25 x 5 = 125 tripleevent possibilities (s_{i}, m_{j} and p_{k}) involving the allocation of, for convenience, 500 counters. All such elicitations would proceed on an iterative basis of changing allocations until the decision maker was content that the display reflected his or her degrees of belief. From Figure 11.5, the elicited probability for the triple event of, e.g., a survival rate of 85 per cent, marketability rate of 80 per cent and market price of $1.50 is 8/500 = 0.016.
FIGURE 11.4  Application of Visualdisplay Procedure for the Elicitation of the Farmer's Subjective Probabilities for the Joint Occurrence of Survival Rate (s_{i}) and Marketability (m_{j}) Levels^{a} in the Spongefarm Example of Table 11.4
Survival rate 
Marketability (m_{j}) 
Total 

(%) 
70 
75 
80 
85 
90 

75 
· 
· · · · 
· · · · 
· 

10 
80

· · 
· · · · 
· · · · · 
· · 
· 
15



· 



85

· · · · 
· · · · · 
· · · · · 
· · · · · 
· · · 
25



· · · 



90

· · · · 
· · · · · 
· · · · · 
· · · · · 
· · · 
40


· · · · · 
· · · · · 
· · · 




· · · · · 



95 
· 
· · · 
· · · · 
· · 

10 
Total 
12 
26 
37 
18 
7 
100 
^{a} The specified levels are the midpoints of exhaustive exclusive intervals centred on the specified levels. Thus a level of, e.g., 80 refers to the range from >77.5 to £ 82.5.
Two features of the above visualdisplay elicitation procedure need to be noted. First, since probabilities are to be derived, the set of events considered must be exhaustive (i.e., cover all possibilities) and exclusive (i.e., not overlap). Second, particularly if joint events are being considered, the number of intervals considered for each variable should be no more than three or four, otherwise the display becomes too difficult to comprehend adequately.
FIGURE 11.5  Secondstage Visual Display for the Elicitation of the Farmer's Subjective Probabilities for the Joint Occurrence of Survival Rate (s_{i}), Marketability (mj) and Market Price (p_{k}) Levels in the Spongefarm Example of Table 11.4
si 
mj 
p_{k} 
Total 

0.50 
1.00 
1.50 
2.00 
2.50 

75 
70 

· 
· · 
· 
· 
5 
80 
70 
· 
· · · 
· · · 
· · 
· 
10 
85

70

· · · 
· · · · · 
· · · · · 
· · 
· · 
20


· · · 




90

70

· · · 
· · · · · 
· · · · · 
· · 
· · 
20


· · 
· 



95 
70 

· · 
· 
· 
· 
5 
75

75

· · · · 
· · · · · 
· · · · 
· · · 
· 
20


· · · 




80

75

· · · · 
· · · · · 
· · · · · 
· · 
· 
20


· · · 




85

75

· · · · · 
· · · · · 
· · · · · 
· · · 
· 
25


· · · · · 
· 



90

75

· · · · · 
· · · · · 
· · · · · 
· · · · · 
· · · 
50

· · · · · 
· · · · · 
· · · · · 



· · 
· · · · · 





· · · · · 




95

75

· · · · 
· · · · · 
· · · 
· · 

15


· 




75

80

· · · 
· · · · · 
· · · · 
· · · 
· · 
20


· · · 




80

80

· · · · · 
· · · · · 
· · · · · 
· · · 
· 
30

· · · · 
· · · · · 





· · 




85

80

· · · · · 
· · · · · 
· · · · · 
· · · · 
· 
40

· · · · · 
· · · · · 
· · · 



· 
· · · · · 





· 




90

80

· · · · · 
· · · · · 
· · · · · 
· · · · · 
· · · · · 
75

· · · · · 
· · · · · 
· · · · · 
· · · · · 
· 

· · · · · 
· · · · · 
· · · · · 
· · 


· · · 
· · · · · 





· · · · 




95

80

· · · · · 
· · · · · 
· · · · 
· · 
· 
20


· · · 




75 
85 
· · 
· · 
· 


5 
80 
85 
· · · 
· · · · 
· 
· 
· 
10 
85

85

· · · · 
· · · · · 
· · · · · 
· · · · · · 
· 
25


· · 
· · 
· 


90

85

· · · · · 
· · · · · 
· · · · · 
· · · · · 
· 
40


· · · · · 
· · · · · 
· · · · · 



· · · 
· 



95 
85 
· 
· · · · · 
· · · 
· 

10 
75 
90 





0 
80 
90 
· 
· · 
· 
· 

5 
85 
90 
· · · 
· · · · · 
· · · · 
· 
· · 
15 
90

90

· · · · 
· · · · · 
· · · 
· 
· 
15


· 




95 
90 





0 
Total out of 500 
105 
185 
112 
68 
30 
500 
Of course, farmers' degrees of belief in uncertain events are not formed in a vacuum. As well as involving intuition and feeling, they are based on whatever information may be available from experience or from others. The incorporation of such information into probability judgements and possible psychological pitfalls in probability assessment are outlined by Anderson, Dillon and Hardaker (1977, Ch. 2) and Hardaker, Huirne and Anderson (1997, Chs 3 and 4). Fuller treatment of probability elicitation is provided by Morgan and Henrion (1992, Chs 6 and 7).
Judgemental fractile method
As noted in relation to stochastic budgeting (Section 11.9), specialist addin software such as @Risk (Winston 1996) is available for use with spreadsheet software to fit a wide range of probability distributions to risky decision variables. This is particularly useful for the fitting of continuous distributions and gives the analyst the opportunity of visually checking the goodness of fit of alternative distributions. The data to which such distributions are fitted may come from visual display procedures as outlined above or be elicited by the judgemental fractile method (Anderson, Dillon and Hardaker 1977, pp. 2325) or be generated by simulation modelling (Section 11.15).
The judgemental fractile method leads directly to the decision maker's cumulative distribution function (CDF) for the risky variable of concern. The farmer would first be asked for the f_{0.0} and f_{1.0} fractiles, i.e., the upper and lower range of the variable. Next, the f_{0.5} fractile value (or median) such that it is equally likely the uncertain variable will be above or below this value is ascertained. Next, elicit f_{0.25} such that it is equally likely the uncertain variable is less than f_{0.25} and between f_{0.25} and f_{0.5} Similarly, find f_{0.75}; then  continuing to halve the fractile intervals  elicit f_{0.125}, f_{0.375}, f_{0.625} and f_{0.875}. A CDF can then be fitted by hand or by personal computer to the set of elicited fractile values. Thus, if the judgemental fractile method had been used, the CDF of Figure 11.2 could have been sketched from elicited fractile values of, say, f_{0.0} = $0, f_{0.125} = $260, f_{0.25} = $410, f_{0.5} = $585, f_{0.75} = $700, f_{0.875} = $765 and f_{1.0} = $960.
Rectangular distribution
The simplest, albeit the roughest, approach to probability elicitation that an analyst can use is to assume that the decision maker's degrees of belief can be specified as either a rectangular distribution or as a triangular distribution. For a discrete variable having n possible values, the rectangular distribution gives each value a probability of 1/n. For a continuous variable with range (a, b), the distribution is rectangular in shape with a uniform probability of 1/(b  a). The rectangular distribution is thus appropriate if all possible values of the risky variable are assumed to be equally likely. This, it may be argued, is the case when the decision maker feels totally uncertain about the likelihood of possible outcomes; equal ignorance about each may then be translated into an assumption of equal probability of occurrence. However, this begs the question of how the range of the distribution can be set.
Triangular distribution
The triangular distribution is a continuous distribution whose specification requires elicitation of only three values of the risky variable  its lowest, highest and most likely values denoted, respectively, by a, b and m. The formula for the triangular distribution for an uncertain variable X is:
f(X) = 2(X  a)/(b  a)(m  a), X £ m
f(X) = 2(b  X)/(b  a)(b  m), X > m
and the formula for its CDF is:
F(X) = (X  a)^{2}/(b  a)(m  a), X £ m
F(X) = 1  [(b  X)^{2}/(b  a)(b  m)], X > m.
The mean E(X) and variance V(X) of the triangular distribution can be found as:
E(X) = (a + m + b)/3
V(X) = [(b  a)^{2 }+ (m  a)(m  b)]/18
Figure 11.6 gives the triangular distribution and its CDF for the net return data of Table 11.5. While this is a singlevariable distribution for possible decision outcomes, it is based on simultaneous variation in four variables influencing outcomes. For convenience, the distributions of Figure 11.6 are in units of a thousand dollars. To fit the probability distribution, the probability of m = 0.784 is calculated as 0.273 using the above equation for f(X) with a = 0.924 and b = 6.413. To fit the cumulative distribution, F(X) is calculated for a number of X values, e.g., for X = 0, F(X) = 0.068; for X = 3, F(X) = 0.718.
Suppose a farmer's utility function for money gains and losses is approximately represented by U(X) = 2.05X  0.01X^{2} for X£ 80, where X is in units of $10. The farmer has to decide whether to spend more on fertilizer for his or her two ha of crop than last season's $40/ha. The profit from fertilizer use depends on the type of season that occurs. The farmer's subjective probabilities for the type of season are given in the payoff matrix of Table 11.8 along with the possible dollar profits for each alternative action and state (i.e., type of season). Since the farmer's utility function is nonlinear and exhibits diminishing marginal utility (i.e., d^{2}U/dX^{2}<0), the monetary payoffs of Table 11.8 are specified on a total crop basis of two ha. Next, using the farmer's utility function (and remembering that X is in units of $10), the utility of each of these total crop payoffs is calculated to give the utility payoff matrix of Table 11.9. The expected utility of each alternative decision is then calculated using the farmer's subjective probabilities. The optimal choice is shown to be to spend $80/ha  this action has the highest expected utility of 8.512. In contrast, in expected monetary terms, the action of spending $120/ha has the highest EMV of $76 (and would be the optimal choice if the farmer were risk neutral and thus had a linear utility function). Substituting the utility value of the optimal risky act into the utility function and solving for X gives the CE of the optimal risky act as $42, i.e., the farmer would be indifferent between a sure prospect of $42 profit and the risky prospect of outcomes from spending $80/ha on fertilizer for his or her two ha of crop. This CE of $42 compares with the EMV of $64 from spending $80/ha on the two ha of crop. The farmer's risk premium in this instance is thus $(64  42) = $22. He or she would be prepared to forgo 22 'expected dollars of profit' to avoid undertaking the risky prospect.
FIGURE 11.6  Triangular Probability Distribution f(X) and Cumulative Distribution Function F(X) for the Net Return Data of Table 11.5
TABLE 11.8  Monetary Payoff Matrix for a Farmer's Fertilizer Decision Problem together with EMV and CE of Each Risky Prospect
Type of season 
Farmer's subjective probability 
Alternative actions 

Spend 
Spend 
Spend 
Spend 

($ profit or loss) 

Poor 
0.1 
160 
240 
320 
400 
Fair 
0.2 
40 
160 
240 
320 
Good 
0.5 
40 
80 
120 
160 
Excellent 
0.2 
240 
400 
480 
480 
EMV 

44 
64 
76^{a} 
72 
CE 

37 
42^{b} 
40 
26 
^{a} Indicating optimal choice if farmer were risk neutral.^{b} Indicating optimal choice for farmer with utility function U(X) = 2.05X  0.01X^{2}, X £ 80, for X in units of $10 gain or loss.
TABLE 11.9  Utility Payoff Matrix for the Farmer's Fertilizer Decision Problem of Table 11.8
Type of season 
Farmer's subjective probability 
Alternative actions 

Spend 
Spend 
Spend 
Spend 

(Utility payoff) 

Poor 
0.1 
3.536 
5.496 
7.584 
9.800 
Fair 
0.2 
1.672 
7.072 
10.992 
15.168 
Good 
0.5 
4.020 
7.880 
11.580 
15.120 
Excellent 
0.2 
8.688 
13.200 
15.072 
15.072 
EV 

7.500 
8.512 
8.076 
5.224 
Comparison of the certainty equivalents of alternative risky choices is probably the most practical guide available for small farmers in their risky decision making (Makeham, Halter and Dillon 1988). For many, it perhaps approaches the implicit intuitive way in which they make their risky choices. In essence, the CE approach involves:
(1) specifying the set of possible alternative decisions/choices/acts;(2) specifying the set of possible outcomes associated with each possible decision;
(3) specifying the subjective probability distribution of each decision's set of possible outcomes;
(4) specifying the CE of each decision's probability distribution of possible outcomes; and
(5) choosing that alternative whose probability distribution has the largest CE.
Such specification and comparison of each alternative's CE may be carried out informally by introspection or more formally using a payoff matrix as in Table 11.8 or a decisiontree format as outlined in Section 11.12 below. Either way, the CE approach has much to recommend it: first, it is reasonable to expect that for any risky prospect he or she may face, the decision maker can specify some equivalent sure prospect or CE; second, the CE approach is logical in that it recognizes and integrates the decision maker's personal probabilities and personal preferences; and, third, it has the attraction of being based on what might be called guided intuition, i.e., intuition guided by a recognition of the importance of personal probability and personal preference in optimal risky choice. Note that, just as with utility appraisal, the CE approach generally needs to be applied on an aggregate rather than a technical unit basis.
The CE approach can be illustrated by reference to the fertilizer decision problem of Table 11.8. Having either explicitly or implicitly specified the payoffs and probabilities associated with each of the four risky prospects, the farmer would establish introspectively his or her CE for each risky prospect and choose that action with the largest CE. Note that, as implied by the definition of a CE, the farmer's ranking of the risky prospects based on certainty equivalence as shown in Table 11.8 agrees with the utilitybased ranking of Table 11.9.
The four essential features of any risky decision problem  i.e., available acts, uncertain events, uncertain payoffs and their subjective probabilities of occurrence  can be specified/displayed/modelled in the form of a decision tree or decision flow diagram. A decision tree displays the choices available, the uncertain events affecting each alternative, the possible outcomes or payoffs from each alternative and their probability of occurrence. Sequences of possible decisions and uncertain events are sketched in timesequence from left to right as successive branches respectively emanating from decision nodes and event nodes on a tree graph. Working backwards from terminal outcomes (i.e., from right to left), certainty equivalents are compared across alternatives to select a sequence of decisions which maximizes the decision maker's expected satisfaction (Anderson, Dillon and Hardaker 1977, pp. 124130; Hardaker, Huirne and Anderson 1997, pp. 107119). As an example, Figure 11.7 shows the risky fertilizer decision problem of Table 11.8 displayed as a decision tree, first in its entirety (tree A) and then in an equivalent foldedback form (tree B) based on the farmer's CE for each risky choice. Again the optimal choice is indicated as spending $80 per ha on fertilizer.
As the above example indicates, a decision tree is a means of operationalizing the certainty equivalent approach (Dillon and Hardaker 1993, pp. 245252). The decision tree (1) forces the decision maker to explicitly consider alternative actions and possible events influencing their outcomes, (2) enables the decision maker to express a complex problem situation as a sequence of decisions, (3) helps the decision maker to quantify the decision process, and (4) facilitates optimal choice based on the comparison of certainty equivalents. Again, as with utility analysis, outcomes should be specified on an aggregate basis and the risky events considered should be exhaustive and mutually exclusive so that their probabilities of occurrence sum to one.
FIGURE 11.7  Risky Decision Problem of Table 11.8 expressed as a Decision Tree
Figure 11.8 shows the decision tree for a more realistic example in which a farmer has to decide whether or not to plant a new rice variety which promises to increase yield but only if fertilizer is applied. The farmer estimates there is a 40 per cent chance that fertilizer will not be available. Growing period is 90 days for the new variety and 120 days for the old one. With the new variety, double cropping becomes feasible. However, double cropping will give a good yield only if the monsoon is not early. Should the monsoon be early, the farmer risks losing not only the second crop but also its planting costs. The farmer believes there is a 50 per cent chance the monsoon will be favourable, i.e., not early. Two decisions  old or new variety? and single or double crop?  and two chance events  fertilizer availability and timing of the monsoon  are involved in this decision problem. As shown in Figure 11.8, it can be depicted as a twostage decision tree to be solved using certainty equivalence. Note that double cropping and timing of the monsoon are irrelevant if the old variety is planted. The possible outcomes for each possible combination of choices are based on budgeting of each possibility.
Analysis of the decision tree of Figure 11.8 proceeds by determining the CE of each terminal risky prospect and then working backwards through the tree. Thus at decision node 4 there are two prospects: the risky prospect ($1 200, $2 000; 0.5, 0.5) and the sure prospect of $1 300. Suppose the farmer has a CE of $1 500 for the risky prospect. This implies that, if at decision node 4, her or his preferred choice would be to double crop. Likewise, suppose the farmer's CE for the risky prospect at event node 7 is $850. This implies that, if at decision node 5, her or his preferred choice would be to single crop with a sure prospect of $900. The decision tree can thus be folded back to the singlestage tree of Figure 11.9 involving choice between just two risky prospects, i.e., ($1 500, $900; 0.6, 0.4) vs ($1 000, $950; 0.6, 0.4), for which suppose the farmer has CEs respectively of $1 150 and $970. His or her optimal choice at decision node 1 is thus to plant the new variety corresponding to a CE of $1 150. Having chosen the new variety and finding fertilizer to be available, the farmer should choose to double crop at decision node 4 of Figure 11.8. But if fertilizer turns out not to be available, he or she should choose to single crop at decision node 5.
The use of decision trees is not without significant difficulty. Attempts at full depiction of risky decision problems may often lead to trees that are a bushy mess and too difficult to comprehend. Artistry is needed to reduce the tree to the essentials of the decision problem. In particular, continuous variables need to be replaced by discrete approximations. Nonetheless, construction of a decision tree can be a valuable exercise. It forces the farm manager to specify all the relevant acts, events, payoffs and probabilities, as well as the sequence in which acts and events occur. Doing this, even without solving the decision tree, can greatly enhance the farmer's understanding of the risks he or she faces.
Specification and appraisal of decision trees can also be facilitated by the use of such personal computer software as DATAä (TreeAge Software Inc. 1996) and PrecisionTreeä (Palisade Corporation 1996).
Also known as risk efficiency analysis and stochastic efficiency analysis, stochastic dominance is a means of comparing alternative risky choices directly on the basis of their outcomes' cumulative probability distributions (Dillon and Anderson 1990, pp. 146153; Dillon and Hardaker 1993, pp. 241253; Hardaker, Huirne and Anderson 1997, pp. 145152). In essence, it is another form of subjective expected utility analysis but one that is not so discriminating between alternative choices as are direct utility comparison, certainty equivalence and decision tree analysis. As in these other utilitybased approaches, stochastic dominance analysis should usually be conducted on the basis of aggregate outcomes, not on a technical unit basis. Stochastic dominance analysis does not necessarily lead to the one best choice. Rather, by comparison of their outcome distributions, stochastic dominance analysis sorts possible risky choices into two groups: first, those that should not be taken because they are dominated by or are less preferred/inferior to a second group which is not dominated. In making this sort into two groups, stochastic dominance analysis relies on general rather than specific features of the decision maker's utility function.
FIGURE 11.8  Example of a Decision Tree for a Risky Decision Problem
The principles of stochastic dominance are illustrated in Figure 11.10. This shows the cumulative probability distributions or cumulative distribution functions (CDFs) for the outcomes of five alternative decisions: a sure prospect A, a risky prospect B with four discrete outcomes and three risky prospects C, D and E with continuous outcomes. The first general feature of utility used in stochastic dominance analysis is that 'more is preferred to less', i.e., the utility function U(X) slopes upward to the right. Considering the CDFs in Figure 11.10, this implies that the possible decisions A, B and C are clearly less preferred to (or dominated by) the alternative decisions D and E. The CDFs for D and E lie everywhere to the right of those for A, B and C so that, at any level of (cumulative) probability, D and E offer better possible outcomes than A, B or C. Decisions/acts/choices A, B and C should therefore not be taken. Rational risky choice lies only between D and E, neither of which dominates the other because their CDFs intersect.
FIGURE 11.9  Folding Back of the Decision Tree of Figure 11.8 on the Basis of Certainty Equivalence
To discriminate further between possible choices, stochastic dominance analysis requires the additional assumption that the decision maker is risk averse. Consider the CDFs for the two undominated decisions D and E in Figure 11.10. Because their CDFs intersect, neither of these decisions can be said to dominate the other on the basis of more being preferred to less. However, because the area a is greater than the area b , decision E would always be preferred to decision D by a riskaverse decision maker. The general rule is that if, moving from left to right, the cumulative difference in area between two CDFs that cross is always positive, then the rightmost CDF at probability zero will always be the preferred choice if the decision maker is risk averse. This rule means that a CDF with a lower tail lying to the left of another distribution can never dominate it. A number of more advanced techniques of stochastic dominance analysis is also available. These achieve greater discriminatory power among alternative decisions by placing bounds on the decision maker's degree of risk aversion (Dillon and Anderson 1990, pp. 150154; Hardaker, Huirne and Anderson 1997, pp. 149153).
FIGURE 11.10  Stochastic Dominance Analysis of Five Alternative Decisions A, B, C, D and E
The attractive features of stochastic dominance analysis are (1) that it does not require knowledge of the decision maker's utility function, (2) that it is based on direct comparisons between full probability distributions of outcomes and (3) that it can be carried out on a personal computer with the use of such addin specialist software as @Risk (Palisade Corporation 1997). The disadvantages are (1) that it requires knowledge of the decision maker's subjective probability distributions for the possible outcomes of all the relevant alternative decisions, (2) that it necessitates pairwise comparisons of risky alternatives, the potential number of comparisons rising exponentially with the number of alternatives, thereby quickly making the assessment burdensome unless use is made of such software as that of Goh et al. (1989); and (3) that it generally does not lead to the one best alternative but rather to some set of undominated alternatives.
Relative to resourcepoor small farmers, stochastic dominance analysis is, except for possible use on a casestudy basis, not directly useful on an individual basis. As with the other techniques of risky decision analysis, there are too many small farmers (each with their unique personal preferences and degrees of belief) to make general use of the technique either feasible or practical. However, some applications oriented to groups of farmers may be worthwhile. Thus stochastic dominance analysis may be used to evaluate new technology, e.g., in plant breeding programs to assess possible new varieties aimed at particular farming systems or agroecological zones. Such analysis would need to be based on probabilities developed by the analyst through discussion with relevant experts.
The various approaches to risky choice outlined above have all been more oriented to partial rather than wholefarm decision making. This does not fit well with the fact that a farm is a system. Partial approaches are likely to miss systemwide implications arising from the interrelatedness of the system. As shown in Chapter 9, mathematical programming provides a means of casting farm planning and management in a wholefarm system context.
A variety of mathematical programming approaches suited to wholefarm planning under risk has been developed (Hazell and Norton 1986, Ch. 5; Hardaker, Pandey and Patten 1991; Hardaker, Huirne and Anderson 1997, Ch. 9). These may be divided into two classes: first and by far the simpler, those oriented to singlestage risky decision situations (as, e.g., depicted in Figure 11.7) with uncertainty only present in activity net returns or gross margin (i.e., in the Z values of Table 9.9); and, second, those oriented to multistage risky decision situations with consequent uncertainty about future inputoutput coefficients and resource constraints. The latter refer to situations involving more than a single actevent sequence so that initial choices (plans) can be revisited (adjusted) by later decisions as relevant uncertain events unfold over time (as, e.g., in Figure 11.8). Such situations are referred to as having embedded risk and are modelled by what are referred to as stochastic programming models, singlestage or nonembedded risk situations are modelled by risk programming models (Hardaker, Huirne and Anderson 1997, Ch.9; Rae 1994, pp. 338348).
Simplified programming and linear programming, as outlined in Chapter 9, are usually based on expected (i.e., mean) rather than sure activity net returns or gross margins. Ideally, the plans they generate would suit riskneutral farmers but not necessarily the riskaverse majority of farmers. A variety of approaches has been developed to overcome this deficiency. First was quadratic risk programming, a form of nonlinear mathematical programming, based on the strong assumption that utility is maximized in terms of the mean and variance of the probability distribution of total net revenue. Such problems are now readily solved using software such as GAMS (Brooke, Kendrick and Meeraus 1992) which incorporates the powerful nonlinear programming package MINOS. Earlier, lack of nonlinear software led to the development of linear approximations to quadratic programming such as MOTAD, Target MOTAD and meanGini programming (Hardaker, Huirne and Anderson 1997, Ch. 9). Today, with the general availability of such software as GAMS, these earlier approaches to risk programming have been superseded by nonlinear mathematical programming models which (1) directly maximize expected utility given knowledge of the decision maker's utility function or (2) maximize expected utility for specified degrees of risk aversion (Patten, Hardaker and Pannell 1988; Hardaker, Huirne and Anderson 1997, Ch. 9). The latter method, known as utilityefficient programming, is obviously more feasible than the former. Both, however, necessitate knowledge of the probability distributions of total net returns from alternative allocations of farmsystem resources. Again, as for the other utilitybased approaches to risky choice discussed earlier in this chapter, these utilitybased risk programming methods are hardly applicable, except on a casestudy basis, to the mass of small farmers. They could be of use to larger commercial farmers able to buy such special planning services.
Little needs to be said about stochastic programming. As noted earlier, it is aimed at risky decision making where resources initially have to be allocated at the first stage of multistage actevent sequences, i.e., when embedded risk is present. Stochastic programming suffers from 'the curse of dimensionality'  the complexity of the decision model increases exponentially with the number of stages and the number of possible outcomes at each stage. With more than three stages and just a few possible outcomes at each stage, solution is difficult (Hardaker, Huirne and Anderson 1997, pp. 196203). So, while the risky decision problems at which stochastic programming is aimed are real and important, in practical terms their complexity as yet defies analytical solution. The smallfarm analyst can forget stochastic programming as a practical guide to realworld farm planning.
11.15.1 Steps in simulation modelling
11.15.2 Example of Monte Carlo simulation
11.15.3 Simulation flowcharts and computers
11.15.4 Other uses of Monte Carlo simulation
Certainty equivalence is probably the most useful formal approach to such risky annual farm decisions as choice of crop variety, planting date, fertilizer rate, pest control measures etc. For longer term investment decisions, Monte Carlo simulation is probably the most useful formal approach. It is applicable to any level of the farm system but is especially appropriate as a guide to (1) the optimal size/level/investment in expensive capital items (e.g., dams, cocoa dryers, irrigation systems) and (2) the likely profitability of important single activities, especially on large monocrop farms and estates. To illustrate its use, Monte Carlo simulation is applied in Section 11.15.2 to appraise the merit of a farmer's possible investment in banana production.
The broad purpose of simulation is to gain 'experience' in the operation of an activity. There are broadly two ways of acquiring such experience. The first is to actually carry out the investment and then sit back and watch what happens to it over the next 10, 20... years as it is subjected to the occurrence of risky weather and other events. Usually, because of cost and the required long time period, this is impractical.
The second way of gaining experience is to build a model which mimics the essential elements of the activity. This is then subjected to the same array of risky events as the real activity would face. Information is thus obtained about the likely performance of the simulated activity. Such a model is a simulator. The use of simulators is now so widespread that little discussion is needed: airplanepilot training simulators; model cars that train adults in driving skills and children in road safety; mock control panels on which technicians learn how to operate a power station  these are now commonplace. Their agricultural equivalents are activity and wholefarm budgets when these are used for ex ante planning purposes; and for more complex activities, extensions of such budgets in the form of mathematical programming and simulation models.
What all simulators have in common is that they provide knowledge at relatively little cost  how to react to an explosion in a power station without actually blowing it up; whether bananas or coconuts would be the best investment without actually planting them. The second thing they have in common is their ability to compress the acquisition of experience into very short periods of time  e.g., in the case of banana growing, only the ten days or so needed to construct and run the model rather than the 15 to 20 or so years of real time required if a field trial were to be conducted.
Simulation models are of two broad kinds: those that ignore risk and those that attempt to incorporate it. The former are concerned with the imitation of some prescribed event or event series that is assumed to have sure consequences. The cardamom investment of Section 10.12 is of this kind: the 'events' are the specified crop production operations and inputs of Table 10.5, and the 'consequences' are the activity net returns of Table 10.7.
The second kind of simulation model is concerned with evaluating the consequences of risky events. Perhaps the simplest example of simulation of this second kind is a farmer's (perhaps introspective) consideration of past rainfall patterns to assess the likely performance of a future crop. For obvious reasons, formal models of this second kind are referred to as stochastic or Monte Carlo models. Such models incorporate probabilities for the occurrence of relevant risky events. By repeated running (i.e., operation) of the model, with each run based on its own sampling from the incorporated probabilities, a sample of simulated outcomes is generated (Anderson, Dillon and Hardaker 1977, pp. 267272; Hardaker, Huirne and Anderson 1997, pp. 4546, 226228; Rae 1994, pp. 300303). Given a sufficient number of runs, an estimate of the probability distribution of possible outcomes (measured on an aggregate rather than on a technical unit basis) can be obtained. This probability distribution can then be compared with those of other alternatives as discussed above in relation to risky choice based on expected utility, certainty equivalence or stochastic dominance analysis.
There are seven general steps in constructing and operating a stochastic simulation model:
1. Define the objectives. This will determine the nature of required output and the form in which it will be most useful.2. Initiate a flowchart sketch. This is a working document showing the interrelationships in the model and is subject to ongoing revision as the model is developed and used.
3. Provide the input. This consists of numerical raw material on which the model is to operate. In the following example, input consists of:
 a base cost/return budget (Table 11.10).
 a set of weather records and associated economic events (Table 11.11).
 a table of subjective probabilities of occurrence of relevant economic events (Table 11.12).
 an identification code applied to the various economic events (Table 11.13).
 a table of random numbers.4. Formulate management and operating rules. These rules or conventions specify (a) the general conditions under which the model is to operate (e.g., the number of simulated periods or years for which it is to generate output), and (b) the way in which the model is to behave in all relevant circumstances.
5. Trial and calibration. Verification of the model necessitates its calibration against some independent data series of relevance. Too, results from the first few runs of a model may not be of acceptable quality. If so, the model will have to be adjusted and refined (primarily through adjusting the operating rules).
6. Operation. Having been calibrated, the model is run repeatedly according to the revised rules and its output results are accepted. The number of runs of the model is determined by considerations of cost and of the precision required in the estimation of the probability distributions of relevant variables.
7. Interpret and extend the output. The results of step 6 might be of direct use in themselves, but often they require further subjective interpretation, elaboration, explanation if they are to be understood by output consumers.
Step 2 above  development of a flowchart  deserves a little further comment. While deciding on the broad configuration of the simulation model (i.e., what output is required? what input is available? what operating rules are necessary?), a parallel step is to develop the interrelationships among these model components by preparing a flowchart. This serves two purposes. Initially it is a working document, a scratch sheet on which specific model components are specified, arranged, rearranged in logical order. When finalized (if indeed it ever can be regarded as 'final', because nearly all models can usually be improved upon and refined), the flowchart becomes the basis on which a computer program can be written (Section 11.15.3). Use of computers in simulation is, however, by no means mandatory; simple problems can be adequately analysed with a hand calculator.
Background
In following sections a simulation model of a banana production activity is constructed and operated for the purpose of deciding whether a farmer should invest in this type of activity. It is assumed the farm is located in an area for which longterm weather records are available. In this area bananas are a generally profitable investment but, because of tropical storms which periodically reduce yields and destroy or damage plantations, the activity is a highrisk one. In any given year there can be wide variations between expected yield, cost and profit levels for a plantation of given age and those results which actually occur because of the effect of storms. Meteorological records of past storms are available. These are used as the basis for stochastic simulation of the results of establishing and operating a banana activity under typical conditions in the area. First, however, it is useful to note the several different economic effects which storms can have on longterm crops.
From the viewpoint of storm damage, multiyear/perennial crops fall into three groups. The first group consists of crops such as banana which might suffer a yieldeffect involving partial or complete loss of current yield, depending on storm intensity. The first costeffect will be in the form of costs of cleaning up the plantation. A second possible costeffect might also arise in the need to completely reestablish/replant a plantation if it has been completely devastated. However, the need to replant also depends on the age of the plantation at the time of storm occurrence: a storm of given intensity might cause rehabilitation costs in a young crop or a different level of reestablishment costs on a planting which is nearing the end of its useful life.
The second group consists of the larger more massive longlife species  mango, jak, coconut, lychee etc. These might suffer partial or complete loss of current output plus complete or partial destruction of the trees themselves. Also, considering the large volume of vegetative material which might have to be removed from the field, the cost of replanting crops of this group after a hurricane might well exceed the cost of their initial establishment if this had been on clean ground.
In the third group, consisting primarily of vine crops grown on artificial or live trellis supports (pepper, vanilla, passionfruit etc.), the first effect again might be loss of current yield. The second effect, complete plant destruction requiring replanting, will seldom occur. But a third effect  trellis destruction  is common and can be economically more serious than the effect of the storm on the plants themselves. This group also includes ratoon crops such as sugarcane. Again there might be loss of current yield, but the main effect will be in the form of delayed harvesting or increased harvesting costs. If the crop has recently been harvested, however, there will often be no yield  or costeffect at all, regardless of storm intensity.
In addition to their direct yield or costeffect on crops, storms might also have indirect costeffects through damage to the environment or general farm infrastructure. In following sections only those direct effects noted for the first group of crops are considered.
Activity time profile and base budget
As previously with the cardamom investment of Section 10.12, the first step in activity evaluation by simulation is to define the time profile of the activity. Bananas are an activity of intermediate lifespan. Previous bush fallow is slashed down and left to decay in place, planting points are cleared and corms are planted. In Year 2 the stand is maintained but no crop taken. Yield begins in Year 3 and continues for a total of four to six years (here four years is assumed). The plants are then slashed down and the land allowed to revert to bush fallow. Meanwhile, another planting is coming into production on a second block. Thus to produce one ha of crop, about two ha are usually operated on average, one under crop and the second under fallow. On a given farm, each individual intermediatelife planting is one phase of a longterm (continuing, indefinite) banana activity.
Here the activity is evaluated over three sixyear crop cycles. These could occur over a maximum of 3 x 6 = 18 years; but if one or more cycles are prematurely terminated because of storm damage, the total threecycle period will be less than 18 years.
The time profile of the activity provides the necessary framework within which to construct the base budget of Table 11.10. These data are selfexplanatory. It is necessary only to note that the specified yields and costs are those that would apply to the activity in the absence of storms. (Prices are assumed constant.) Subsequent operations of the simulation model discussed below are concerned largely with making adjustments to these base yield and cost data to allow for the random effects of storms of varying levels of intensity.
TABLE 11.10  Base Budget for Simulation Analysis of Banana Investment (per Ha Basis)
Measure 
Year of crop cycle 

1 
2 
3 
4 
5 
6 

Yield (cases) 
0 
0 
150 
250 
200 
100 
Price ($) 
2 
2 
2 
2 
2 
2 
Gross return ($) 
0 
0 
300 
500 
400 
200 
Total cost ($) 
160 
80 
60 
60 
40 
20 
Net return ($) 
160 
80 
240 
440 
360 
180 
Storm incidence and effects
The basebudget data of Table 11.10 must now be adjusted according to possible storm damage in any cropcycle year. The first step is to quantify these future possible storm events. The best guide to what to expect of future weather is what has happened in the past complemented by any acceptable forecast information that may be available. This does not mean that the past sequence of weather events and intervals between them is expected to be exactly duplicated in the future. Rather, it is to say that if three hurricanes have occurred in the past 100 years, it is a reasonable subjective judgement to expect three more in the course of the next 100 years. The particular future years in which they will occur cannot be forecast, nor their intensity, nor the sequence in which future storms of varying intensity  hurricanes, gales, storms, 60knot winds etc.  will arrive.
Storm records for the area extend back to 1909. These are shown in the first two columns of Table 11.11 for the 87year period to 1995. On the assumption that this past pattern of weather events will occur again (but only in some unknown random order), it is now necessary to quantify the economic effects of storms of various intensity levels. As tabulated in Table 11.11, in order of decreasing strength there are five types of storm: hurricanes (or tropical cyclones or typhoons), severe gales, gales, moderate gales and storms. At this point, the objective is to determine the effect of weather events of each of these intensity levels on the economics of banana production.
In the present example, no recorded information on the relation between storm intensity and damage is available. Bananas in the area are a smallholder crop and growers do not keep production records for more than a few years. So several operations towards defining some working relationship between storm intensity and economic loss are necessary. First one would try to establish some informal correlation. Although formal production records are not kept, the memories of some individual growers, older family members and the villages collectively extend back many years. Armed with the weather record of Table 11.11, the analyst would go into the villages and ask two kinds of questions about the effect of storms on banana yields and costs: 'What did happen ... (to past crops as a result of past individual storms?)'; and 'What do you think would happen ... (if a hurricane, severe gale, gale etc. were to hit the plantations today?)'.
After enough farmers in enough villages  say around 50 farmers in all  had provided their subjective recollections and estimates, a schedule relating storms of the five intensity levels to their respective yield and costeffect on bananas could be constructed. Ideally this would be in percentage terms: e.g., a past hurricane or severe gale would have caused a decrease in yield (and gross return) of 50 per cent (relative to what it otherwise would have been as per Table 11.10) and a cost increase of 100 per cent of the initial establishment cost of $160 per ha (Table 11.10); a moderate gale would have reduced crop income by 30 per cent and increased cost by 60 per cent of initial establishment costs etc.
These percentage effects of past storms of several intensity levels are tabulated in the two rightside columns of Table 11.11. However, when they are applied to future storms the magnitude of effect is also partly determined by the place of that storm in the storm sequence (whether a strong gale follows a hurricane etc.) and by the age of the plantation at the time of storm occurrence. The various situations are provided for below in the formulation of model operating rules.
TABLE 11.11  Incidence and Effect of Major Weather Events on Banana Crops in the Study Area, 1909 to 1995
Weather record 
Effect 

Year 
Type of event 
Yield loss^{a} 
Plant damage^{b} 
(%) 
(%) 

1909 
Hurricane 
50 
100 
1911 
Severe gale 
50 
100 
1913 
Severe gale 
50 
100 
1914 
Moderate gale 
30 
60 
1917 
Severe gale 
50 
100 
1918 
Gale 
40 
80 
1924 
Hurricane 
50 
100 
1925 
Moderate gale 
30 
60 
1927 
Storm 
20 
40 
1928 
Hurricane 
50 
100 
1932 
Storm 
20 
40 
1933 
Gale 
40 
80 
1935 
Storm 
20 
40 
1936 
Severe gale 
50 
100 
1939 
Hurricane 
50 
100 
1943 
Moderate gale 
30 
60 
1944 
Severe gale 
50 
100 
1948 
Severe gale 
50 
100 
1951 
Severe gale 
50 
100 
1955 
Gale 
40 
80 
1961 
Moderate gale 
30 
60 
1964 
Gale 
40 
80 
1965 
Moderate gale 
30 
60 
1966 
Gale 
40 
80 
1967 
Gale 
40 
80 
1971 
Gale 
40 
80 
1977 
Moderate gale 
30 
60 
1982 
Severe gale 
50 
100 
1988 
Hurricane 
50 
100 
1990 
Hurricane 
50 
100 
1993 
Severe gale 
50 
100 
^{a} Measured as percentage of basebudget yield (Table 11.10). Translates directly to loss in gross revenue.^{b} Translates into a cost increase of this percentage of initial plantation establishment cost of $160 per ha.
Probability of storm losses and their Monte Carlo sampling
Having developed the relationship between storms of the five intensity levels and their economic consequences, the next step is to arrange these latter data as a probability distribution. This is done in Table 11.12. Storms which result in a 50 per cent yield loss and a 100 per cent damage level are given an event identification number of I; those which result in 40 per cent yield loss and 80 per cent damage are given an event identification number of II etc. (This identification system refers to the level of economic effect, not necessarily to the level of storm intensity because, as noted above, the level of storm effect will depend partly on time of occurrence of the storm in relation to the bananaage cycle (see Operating Rules below).)
TABLE 11.12  Probability of Storm Losses in Banana Production in the Study Area
Aspect 
Type of storm 

I 
II 
III 
IV 
V 

Yield loss (%) 
50 
40 
30 
20 
0 
Plant damage (%) 
100 
80 
60 
40 
0 
Frequency in 87 years 
15 
7 
6 
3 
56 
Probability of occurrence 
0.17 
0.08 
0.07 
0.03 
0.65 
Based on Table 11.11, the data of Table 11.12 indicate that economicloss levels of 100 and 50 per cent (damage costs and fruit loss, respectively) occurred in 15 of the 87 years on record. They have occurred with a relative frequency of 15/87 = 0.17 and can be expected to occur again in any future year with a probability of 0.17. The probabilities of occurrence of the other categories of progressively less serious losses/costs are 0.08, 0.07 and 0.03, and there is a 0.65 probability that in any future year no economic losses of either kind will occur. Though based solely on historical data, these probabilities are subjective because they are accepted by the decision maker, i.e., he or she takes them as his or her degrees of belief in storm occurrence.
Having Table 11.12, the next step is to construct Table 11.13 which assigns an identification code to each of the five possible economicloss events I to V of Table 11.12, based on the probability of occurrence of each respective event. Since an event of type I is expected to occur in 17 per cent of future years, it is assigned the set of 17 code numbers 0 to 16 within the set of numbers 0 to 99; an event of type II with a probability of occurrence of 0.08 is given the set of eight code numbers 17 to 24 etc. as shown in Table 11.13.
The last piece of input needed in model construction is a set of random numbers (as found in many introductory statistical texts or as generated by computer) containing values over at least the same range as the values chosen for the event identification code, i.e., 0 to 99 in the present instance. Part of such a randomnumber set might be the randomly chosen sequence:
46 
93 
38 
59 
16 
4 
39 
74 
69 
34 
67 
27 
31 
19 
98 
41 
37 
9 
21 
81 
40 
87 
76 
8 
72 
TABLE 11.13  Assignment of Event Identification Code Numbers to Economicloss Events in Banana Investment Simulation Exercise
Economicloss event 
Code numbers 

Event 
Yield loss 
Plant damage 

(%) 
(%) 

I 
50 
100 
0 to 16 
II 
40 
80 
17 to 24 
III 
30 
60 
25 to 31 
IV 
20 
40 
32 to 34 
V 
0 
0 
35 to 99 
A run of such random numbers is used to generate the risky event sequences to which the model will be subjected. Thus, referring to the identification code numbers specified in Table 11.13, the number 46 indicates the occurrence of an economicloss event of type V, 93 indicates the same, likewise for 38 and 59, while 16 indicates an economicloss event of type I. This procedure is referred to as Monte Carlo sampling. It is the mechanism for making the simulation stochastic (Dillon and Hardaker 1993, Section 5.5).
This concludes the third step in model construction, preparation of input, as listed in Section 11.15.1. The next step is the formulation of operating rules.
Operating rules for the simulation model
The second part of a simulation model consists of operating rules or conventions which govern the behaviour of the model under all relevant circumstances (here under all possible relevant weather conditions). The rules formulated for the banana example are listed and discussed briefly below. Rules 1 to 3 are general assumptions relating to how the model is to be managed, e.g., the timeperiod over which it is to be run. Rules 4 to 10 are technical conventions which, as noted earlier, have been developed largely on the basis of what smallholder growers in the area collectively think would happen to yields and costs as a result of possible storm events occurring over a crop cycle, supported by consideration of the effects of past storms.
Rule 1 
Evaluation period. The banana activity will be evaluated over three cycles. In the absence of storms this implies a total period of 18 years. 
Rule 2 
Cycle length. Each normal cropping cycle will be over six years; however, if or when severe 100 per cent or 80 per cent plant damage occurs in Year 5 of any cycle, that cycle will be terminated at that point. (Storm severity levels are discussed in Rules 8, 9 and 10 below and, for historical storms, were listed in Table 11.11). 
Rule 3 
Each cycle after the first will start in the year following the end or termination of the previous cycle. Thus Year 1 of the second cycle will occur in Year 7 of the activity, assuming the first cycle has run its normal sixyear course. (This will result in production 'gaps' between cycles but farmers have other income sources such as coconuts, jak etc. The assumption simplifies discussion. The alternative which many farmers do in fact adopt is to operate three rather than two fields of bananas.) 
Rule 4 
The economic effects of all storms of whatever intensity will be in terms of effects on returns and costs as given in Table 11.12. 
Rule 5 
In the absence of storms the annual net return of the activity will be as shown in the base budget of Table 11.10. 
Rule 6 
Plant damage caused by a storm will be repaired in the year of storm occurrence. The cost of this will be proportional to the severity of the storm and based on the cost of establishing the plantation in Cycle Year 1 ($160). Thus a storm causing 100 per cent damage will increase activity costs in the year of occurrence by $160, a 20 per cent damage storm by $32 etc. 
Rule 7 
The costs of storm damage repair will be in addition to the normal operating costs of the relevant year as listed in the base budget of Table 11.10, i.e., total costs in any year will consist of normal operating/maintenance costs plus damage repair costs (if any). 
Rule 8 
A severe storm causing 50 per cent or 40 per cent yield loss in any year will result also in 25 per cent or 20 per cent yield loss, respectively, in the following year if it is storm free; less severe storms will cause yield loss only in the year of storm occurrence. 
Rule 9 
If a storm causes 100 per cent or 80 per cent plant damage in Cycle Year 5, that field will be abandoned and a new planting will be commenced (in a second field) in the following year. This was noted in Rule 2 above. 
Rule 10 
If a storm of any intensity occurs in Year 1 of any cycle, it will have no economic consequences. (Newly planted banana palms protected in deep planting holes are stormimmune.) 
At this point these operating rules are tentative. The model would now be subjected to a few trial runs. Depending on the acceptability of results, the rules might be retained, adjusted or perhaps other rules added. Discussion of this calibration step is here deferred to a later section.
Operation of the model
The model is 'run' by taking the gross return/total cost/net return lines of the base budget of Table 11.10 and successively adjusting these activity 'results' according to the occurrence of stormcaused economicloss events (Table 11.12). The occurrence of these events is determined by (successive) selections from a set of random numbers, each of which corresponds to the identification code of one of the five possible risky events (Table 11.13).
Since the evaluation is to be over three consecutive cycles of crop, it is convenient to refer to model operation over these three cycles as one 'run'. The results of the first run (of three crop cycles) are tabulated in the worksheet of Table 11.14. The starting point for Run 1 is provided by the base budget's gross return/total cost/net return data of Table 11.10. 'Activity Year' refers to the point in time occupied by a production year, starting with the first year of the first cycle and continuing to the last year of the last cycle. (Some cycles might not complete their normal sixyear term.) 'Cycle Year' refers to the production year within each successive crop cycle.
TABLE 11.14  Worksheet for a Run of the Banana Monte Carlo Simulation Model (per Ha Basis)
Cycle Year: 
1 
2 
3 
4 
5 
6  
Base budget: 





 

Gross return ($) 
0 
0 
300 
500 
400 
200 

Cost ($) 
160 
80 
60 
60 
40 
20 

Net return ($) 
160 
80 
240 
440 
360 
180 
Run1/Crop Cycle 1/Field 1  
Activity year: 
1 
2 
3 
4 
5 
6  
Random number: 
(46) 
(93) 
(38) 
(59) 
(16) 
Stop  
Economic loss: 





 

Gross return (%) 
0 
0 
0 
0 
50 


Cost (%) 
0 
0 
0 
0 
0 

Simulated outcome: 





 

Gross return ($) 
0 
0 
300 
500 
200 


Cost ($) 
160 
80 
60 
60 
40 


Net return ($) 
160 
80 
240 
440 
160 

Run1/Crop Cycle 2/Field 2  
Activity year: 
6 
7 
8 
9 
10 
11  
Random number: 
(4) 
(39) 
(74) 
(69) 
(34) 
(67)  
Economic loss: 





 

Gross return (%) 
0 
0 
0 
0 
20 
0 

Cost (%) 
0 
0 
0 
0 
+40 
0 
Simulated outcome: 





 

Gross return ($) 
0 
0 
300 
500 
320 
200 

Cost ($) 
160 
80 
60 
60 
104 
20 

Net return ($) 
160 
80 
240 
440 
216 
180 
Run1/Crop Cycle 3/Field 1  
Activity year: 
12 
13 
14 
15 
16 
17  
Random number: 
(27) 
(31) 
(19) 
(98) 
(41) 
(37)  
Economic loss: 





 

Gross return (%) 
0 
30 
40 
20 
0 
0 

Cost (%) 
0 
+60 
+80 
0 
0 
0 
Simulated outcome: 





 

Gross return ($) 
0 
0 
180 
400 
400 
200 

Cost ($) 
160 
176 
188 
60 
40 
20 

Net return ($) 
160 
176 
8 
340 
360 
180 
As a starting point, the gross return/total cost/net return lines of the base budget of Table 11.10 are reproduced at the top of the worksheet of Table 11.14. The next step is to select a random number for Year 1. It is 46. From Table 11.13 this number falls within the event identification code range 35 to 99, signifying the occurrence of event V, i.e., a year with no storm. (In any case, even had a severe storm occurred in Year 1, it would have been ignored according to Rule 10.) Since there is no storm effect in Year 1, the simulated outcome is the same as that of the base budget, i.e., a net return of $160. Analysis then moves on to Year 2.
A second random number is selected, 93, again signifying a stormfree year. This is repeated in Years 3 and 4; but in Year 5 the random number 16 brings with it a hurricane or strong gale. From Table 11.12 this event will decrease current gross return (through fruit loss) by 50 per cent and also, by Rule 6, increase cost by 100 per cent of initial plantation establishment cost, $160. However, according to Rule 9, since the field is nearing the end of its useful life, it would instead be abandoned. It will therefore not incur cleanup costs. The only economic loss due to this hurricane will be loss of 50 per cent of the yield/gross return. A new field will be started the following year, i.e., in Activity Year 6 (Rule 2).
Continuing to Year 1 of the new crop cycle (Activity Year 6), the random number of four again signals a major storm; but according to Rule 10 this causes no loss. The following Cycle Years 2, 3, 4 are storm free. Then in Cycle Year 5 the random number 34 signifies an economicloss event of type IV causing a gross return decrease of 20 per cent and a cost increase of 40 per cent. Since this is a minor rather than a major storm (Rule 9), the field will be cleaned up rather than abandoned. Gross return this year will be $400  0.2($400) = $320, while total cost will be the normal annual current cost of $40 plus 40 per cent of the initial establishment cost of $160, i.e., a total of $104. Net return is thus $216. The following year, Cycle Year 6, is stormfree (random number 67); and there is no carryover damage from the storm of the previous year (Rule 8) because that would have been incurred only by a type I or II storm.
Continuing, Year 1 of the next (third) cycle, although suffering a type III storm, incurs no economic loss by virtue of Rule 10. However, a similar storm in Cycle Year 2 does cause an increase in cost. In Cycle Year 3 a type II storm occurs which has direct effects on returns and costs. However, according to Rule 8, it also has the secondary effect of reducing yield/gross return in the following year, which would otherwise have been $500, by 20 per cent, i.e., to $400. Except for this effect in Cycle Year 4, the rest of the crop cycle is stormfree. This concludes Run 1 of the Monte Carlo simulation.
Economic evaluation of Run I
The simulated results of operating the activity over three cycles for a total of 17 years constitute an irregular stream of annual net returns (some of which are negative). For purposes of comparison with other possible activities/investments, it is convenient to convert this stream to a present value (PV) as per Section 10.7. As calculated in Table 11.14, the stream of net returns from Run 1 of the model is $160, 80, 240, 440..., 360, 180 occurring in sequence over Activity Years 1 to 17. Assuming an annual discount rate of ten per cent, this stream has a PV of $773. On an amortized annual basis, this is equivalent to an annuity of $96 per year over the threecycle or 17year evaluation period (Section 10.10).
The particular result of Run 1 is in a sense accidental: it occurred as a result of the 'accident' of selecting the particular sequence of random numbers 46, 93, 38... 37 from the table of random numbers. If the table of random numbers had been entered at any other point, a quite different series would have been selected. This would have generated a different weather experience and thus a different final result. In other words, operating the model for only a single run does not give sufficient experience to claim with reasonable confidence that net returns from this banana activity over three cycles would have a PV of $773. More experience is needed.
The necessary additional experience would be gained simply by running the model repeatedly until sufficient estimates of the activity's PV of net returns were available to give a reasonable estimate of its probability distribution. With the aid of a computer (which can do nothing that an analyst can't do but does it faster), any number of runs can be made  dozens or hundreds as need be  and any amount of simulated banana growing experiences can be accumulated. For illustrative purposes, only 20 runs of the model are used here. This is probably about the minimum number of runs on which to attempt judgement of the banana activity. Fifty to 100 runs would be far better.
Economic evaluation of all runs
As noted in Section 11.15.1, the final step in simulation modelling consists of interpreting the results. With 20 runs of the banana model, the results in terms of the PV of each run (assuming a discount rate of ten per cent) might be as follows: Run 1: $773; Run 2: $516; Run 3: $810; Run 4: $498;... Run 20: $382. Having these results, there are various approaches that might be taken to assess them from the decisionmaking perspective of whether to invest in bananas or in some other alternative (for which analogous information would presumably be available).
One obvious possibility is to take the average of this set of PVs, $529, and use this  after conversion from its technical unit to an aggregate basis  as the basis of comparison. However, use of this single value might imply a degree of certainty which  considering the nature of the data and the many operating assumptions which have been made  is not warranted. Alternatively, the mean might be paired with the standard deviation of the set of PVs to provide a better basis for the comparison of risky alternatives.
However, considering the risk aspect, a better approach might be to consider the returns from banana growing in terms of reliability rather than in terms of averages. To this end the simulated results can be arranged as a reverse cumulative probability distribution (i.e., reverse CDF) as shown in Figure 11.11. This is obtained by graphing the estimated PVs from the 20 simulation runs against their respective reverse cumulative probabilities. These latter are developed in the worksheet of Table 11.15 based on the sparsedata rule that, given m observations on a continuous random variable, when these are arrayed in ascending order of size, the nth observation is a reasonable estimate of the n/(m+1) fractile (Anderson, Dillon and Hardaker 1977, pp. 4244). Given these fractiles the CDF of PV (i.e., F(PV)) can be specified and the reverse CDF values 1  F(PV) calculated as in Table 11.15. A freehand curve of the relationship between the simulated PV values and their associated reverse cumulative probabilities can then be sketched as in Figure 11.11. This is an incomereliability schedule. It shows, for any level of PV, the reliability of getting this or a higher level of return from three sixyear cycles of banana production, e.g., from Figure 11.11, a return of $600 has a reliability of about 30 per cent. Conversely, reading up from the reliability level axis indicates the minimum level of return which can be expected at any particular level of reliability. It is apparent that high returns are associated with low reliability. Thus a PV of $820 or greater would be achieved only about five per cent of the time. The other 95 per cent would result in a return of less than $820.
TABLE 11.15  Estimated CDF and Reliability (Reverse CDF) of PV of Net Returns from Monte Carlo Simulation of Banana Production (per Ha Basis for Three Sixyear Cycles)
Run no. 
PV^{a} 
Cumulative probability^{b} 
Reliability 
(n) 
F(PV) 
[1F(PV)]100 


$ 

% 
1 
308 
0.048 
95.2 
2 
331 
0.095 
90.5 
3 
353 
0.143 
85.7 
4 
382 
0.191 
80.9 
5 
420 
0.238 
76.2 
6 
435 
0.286 
71.4 
7 
467 
0.333 
66.7 
8 
472 
0.381 
61.9 
9 
477 
0.429 
57.1 
10 
498 
0.476 
52.4 
11 
516 
0.524 
47.6 
12 
558 
0.571 
42.9 
13 
563 
0.619 
38.1 
14 
572 
0.667 
33.3 
15 
621 
0.714 
28.6 
16 
645 
0.762 
23.8 
17 
676 
0.809 
19.1 
18 
702 
0.857 
14.3 
19 
773 
0.905 
9.5 
20 
810 
0.952 
4.8 
^{a} Present value (at an annual discount rate of ten per cent) of the stream of annual net returns generated by each run of the simulation model. Listed in ascending order of size.^{b} Calculated as n/(m + 1) for m = 20.
An incomereliability schedule such as that of Figure 11.11 might be used as follows. The farmer or analyst would specify the level of reliability he or she demanded of the activity before undertaking to invest in it. This might be 80 per cent. He or she would then use the curve to read off the minimum PV of net returns per ha to be expected with this level of reliability. At 80 per cent this is about $400. If this particular returnreliability combination is satisfactory in the eyes of the farmer he or she would go ahead with the banana activity; if it is not, then he or she would seek some alternative investment.
FIGURE 11.11  Estimated Reliability of the PV of Net Returns from Banana Production
Such a return vs reliability approach is obviously practical. It is also attractive in its direct recognition of tradeoffs between income and risk. Nonetheless, it has the disadvantages (1) of involving an arbitrary level of reliability and (2) of not assessing the probability distribution of outcomes in its entirety. A better approach would be to use one of those based on subjective expected utility. In particular, the decision maker might determine his or her CE for the probability distribution of the PV of aggregate net returns from investing in bananas and compare this CE with those of the alternative choices. Alternatively, stochastic dominance analysis as per Section 11.13 might be used to guide choice  though, unlike the CE approach, this would not necessarily determine the optimal choice.
Stochastic dominance analysis of the PVs of the simulated net return streams of alternative longterm risky investments is greatly facilitated by such personal computer software as DATA (TreeAge Software Inc. 1996) and @Risk (Palisade 1997). With a sufficient number of runs (say 250 or so) of the simulation model, such software can be used to generate a cumulative frequency distribution of the outcomes from each alternative. Again these outcomes should be measured on an aggregate rather than technical unit basis. Examples are provided by Hardaker, Huirne and Anderson (1997, pp. 226228), Rae (1994, pp. 301303) and Winston (1996). Such computergenerated CDFs can then be compared by stochastic dominance analysis as illustrated relative to Figure 11.10, or by certainty equivalence, or by direct utility evaluation if the decision maker's utility function is known. However, as noted with the other formal approaches to risky farm decisions, such approaches to the assessment of the outcomes from stochastic simulation are probably only relevant to investment appraisal on large farms and estates. Except perhaps for representativefarm analysis and case studies, they are not justified for small farms.
Model calibration
An important step in model construction is calibration or verification to ensure that results can be used with a reasonable degree of confidence. Calibration is achieved by comparing the output of trial runs (e.g., the simulated net return data of Table 11.14) with some independent data series and, if necessary, making adjustments to the operating rules so as to improve the model's performance as a mimic of reality. Either the final output of the model or some intermediate output may be used as the basis for calibration.
Calibration of some models is easy. Models designed to simulate the operation of a farm irrigation supply dam in which the uncertain parameters are rainfall/runoff/inflow to the dam and irrigation outflow from the dam can often be accurately calibrated, in regard to inflow, by reference to the behaviour of a local municipal reservoir for which good records are kept. On the other hand, calibration of outflow (determined by rainfall and crop requirements) will usually be more difficult.
For the banana model, several calibration possibilities might exist. Thus, although the model relates to a particular area, it might be possible to find a farm or research station either nearby or elsewhere which has a similar weather regime and for which crop production records are available. A second possibility might be to base calibration on the relative behaviour of some different crop which had been subject to the same weather events as the subject banana crop. Thus there might be coconut estates in the area which have production/marketing records extending back the required number of years. To use such records for calibration it would only be necessary to establish some linkage, in terms of the effects of hurricanes, gales etc., between coconuts and bananas.
The third possibility could be to seek some other relevant data series for the same crop. For the banana model, such a series might be, e.g., the annual total of banana exports in cases from the region. Suppose such a data series is available for the 32 years 1964 to 1995. Because this is a series of physical quantities, the model's intermediate output in terms of percentage fruit loss due to storms  lines 1 and 2 of Table 11.12  provides a suitable calibration factor (and the final output of net returns is, for this purpose, ignored). Also, for calibration purposes the model is assessed using the actual weatherevent sequence over the period 1964 to 1995 for which the calibration series, exports, is available (rather than using randomly generated weather events as in Table 11.14). There should  it is hoped  be some reasonable correlation between the physical yieldeffects of storms at farm level and the calibration series (exports) at regional level. (However, because the simulated outputs refer to production on an individual farm while the calibration series refers to exports from all farms in different localities in the region  and all localities are not affected to the same extent by a given storm  a very close correlation could not be expected.) In Figure 11.12, the level of exports in storm years is plotted against the simulated farmyield decline in those same years. The graph indicates, as hoped, that there is a strong correlation between the level of exports and storminduced decreases in simulated farm yield. The stochastic simulation model, at least in these terms, may be accepted as well verified. Had the analysis of Figure 11.12 not proved satisfactory, some adjustments either to the model's operating rules or to the storm effects of Table 11.12 would be necessary.
In summary, calibration is an essential part of simulation model building. It proceeds by comparing a sample of results from the model with some independent data series relating to the events being simulated. Trialanderror adjustments are then made to the model's assumptions and operating rules until the model is believed to be generating reliable output. In the final analysis, this inevitably involves subjective judgement.
As noted previously, preparation of a flowchart is always a useful early step in model construction and in more complex problems an essential one. A flowchart for the banana example would be developed along the lines begun in Figure 11.13. This serves two main purposes. First, problem specification and analysis are always facilitated by a flowchart. Even the attempt to construct one will compel the analyst to think more carefully about what initial input data are needed, what operating rules are to be applied to the data, and the internal interdependence between output from one part of the model and its use as input to another. Second, a flowchart greatly facilitates communication about the model both with other experts and with its intended beneficiaries. Thus, while it is not necessary that the farm management analyst be able to write a computer program, it is essential that he or she be able to communicate with the programmer. The best way to facilitate this is via a flowchart. It is also now possible, using such software packages as Stella II (High Performance Systems Inc. 1997), to develop Monte Carlo simulation model flowcharts dynamically on screen.
Whether the model should be computerized or the calculations done by hand depends on several cost/benefit factors in addition to the obvious factors of problem complexity and computer availability. The banana model could probably be constructed and run the required number of times using a hand calculator in about a week. It would take about twice as long to obtain results if a computer program has to be written. Therefore, if the model is to be used only once and then discarded, or applied to only one farmer or a small group of farmers, it would hardly justify computerization.
Essentially, whether computerization is justified will depend on the size of the market for model output or the number of potential users. This can often be increased 'horizontally' by so generalizing the model that its output is applicable to a maximum number of farmers. Simply put, a government agriculture department is more likely to fund a model which is applicable to all banana growers in an area than one intended to benefit only one or a few farmers; and a regional development agency is more likely to fund a general model which is applicable across its region than one which is applicable only to part of the region. Likewise, a model which is sufficiently general to handle a number of crops subject to the same weather events, e.g., bananas, coffee, coconuts etc., would justify computerization more than would a cropspecific model.
The number of potential users may also be increased by 'vertical' generalization: banks, wholesalers, export shipping companies etc. are all firms whose operations will be subject in greater or lesser degree to the same weather events as affect farmers. Therefore, when feasible, it is desirable to construct a model which will meet the needs both of farmers and of these various farmrelated agencies.
FIGURE 11.12  Calibration of Banana Simulation Model by Plot of Simulated Farmyield Decreases against Regional Exports of Bananas, 1964 to 1995
A final point is that, although computer use is increasingly widespread (particularly in risky decision analysis and simulation using such software packages as DATA, @Risk and Stella II), there still remains an aura of sanctity surrounding computergenerated output. This often obscures the fact that such output is only as reliable as is the quality of input data and the logic of the operating rules by which this input has been processed. If there are deficiencies in the input or illogicalities in the rules, these will not be removed by using a computer any more than they would be by ciphering with one's fingers instead of with clam shells. The risk of accepting fiction for fact merely because results are obtained with a computer is greatest in those models (including most simulation models) which contain a high degree of subjectivity in their composition. It is particularly great where there exists no really reliable independent data series against which model operation can be calibrated.
FIGURE 11.13  Initial Elements of a Flowchart for the Monte Carlo Simulation of Banana Production
In addition to its use in planning longterm crops, Monte Carlo simulation has many other possible applications in farmsystem planning and management. Some of these are briefly noted below.
Application to production activities
Haymaking. This activity usually requires a minimum period of five to six days of hot, fine, lowhumidity weather. If these conditions do not prevail after the alfalfa or berseem has been cut, the crop (or at least the current harvest if it is a semipermanent species) will be lost. Typically, there will be only some periods of the year when the right weather conditions will or will not exist with certainty. There will be transitional periods when successful haymaking is a matter of chance. Simulation of a haymaking activity based on daily weather records would determine the number of successful cuts that can be anticipated over the year or, alternatively, the reliability that may be attached to obtaining five, six, ... successful cuts, and thus the probable income from this activity.
Crop systems. A common planning problem in monsoon lands is to determine (a) the feasibility of continuous paddy under given 'natural' conditions; or (b) the ways and extent by which these conditions must be modified (e.g., by artificial graindrying) in order for continuous cropping to become possible; or (c) that particular crop sequence or rotation which is both feasible and economically optimal. Here the common physical constraint is usually rainfall or irrigation water supply. The farming system, income, employment  indeed the whole village economy  all depend on the arrival date, intensity and duration of the annual monsoon. These are highly random events. In longsettled areas the relative feasibility and comparative economics of various traditional cropping/farm systems will usually have been determined long ago from centuries of experience. But when new crops/varieties become available and in new settlement areas where experience is lacking, economic information can often be generated by weatherbased simulation.
Technology evaluation. It is a truism that not all 'improved' technologies are really helpful when applied at village level. Many of them, whether of the biological or mechanical kind, promise much but often deliver little. These 'empty boxes' can often be identified ex ante by testing them as components of simulated systems. E.g., tractors often appear to be the best means of achieving production intensification, or of ensuring that one critical crop can be produced within some limited time period. But if the weather and soil moisture conditions which tractors require are carefully specified, it might be found (by simulation) that these necessary conditions occur only infrequently, and gains in the speed of performance of operations are more than offset by time spent in waiting for the necessary conditions to occur. Slowmoving buffalo might in fact be the fastest method of land preparation when their use is simulated within the context of a specific weathersoilseconomic environment.
Farm development through intensification. Production intensification is generally sought by maximizing the number of crops produced on a given land area in a given time, by selecting crop species/varieties which have relatively short growing periods and by selecting production technologies which minimize the interval between successive crops/phases. The general planning problem is illustrated by the flowchart of Figure 11.14, the core of which shows the time taken for each operation in producing one cycle of a relayplanted paddy crop using alternative technologies under a range of operating assumptions. The average time required for each operation (days of time, not number of labour days) is shown and those operations that are weatherdependent are marked by an asterisk. Actual values of the latter would be generated by running the model for simulated series of weather events (rainfall amount, duration, humidity level etc.). (Each operation/technology would, of course, have associated inputs/costs and outputs/returns; these are not shown.) Because of the number of possible technologies at each weatherdependent production stage, there are literally thousands of routes through the model; computer simulation is the only feasible way of evaluating them.
Application to resourcegenerating activities
Discussion so far has concerned the simulation of production activities. As suggested by the following examples, there is also much scope for the use of simulation techniques in the evaluation of resourcegenerating activities within farm systems.
Farm dams and village watersupply ponds. Here the usual problem is to determine the optimal size/capacity/cost of these structures. Operationally, the procedure consists of reconciling by simulation some set of daminflow factors (rainfall, runoff) with some set of damoutflow factors (seepage loss, evaporation loss, irrigation pumping) in a way which maximizes benefits relative to costs. This particular problem is usually handled by successively specifying several dam sizes/capacities, applying water from these alternative storage situations to alternative possible crops or crop sequences (which will generally have varying water requirements), and then extending such application successively to some increasing size of irrigated area. As an example, five possible sizes of dam might be specified, along with five possible crops or crop rotations to be grown on five possible total areas of irrigated land:
Dam capacity (10 000 m^{3}): 
1 
2 
3 
4 
5 
Crop sequence*: 
PPP 
PPM 
PMM 
PMS 
MSS 
Irrigated area (ha): 
5 
10 
15 
20 
25 
* P paddy; M maize; S soybean.
The 5 x 5 x 5 = 125 combinations of dam size/crop/irrigated area, each of which represents one possible activity, would then be evaluated by simulation in which, say, 20 years of historical daily rainfall records would provide the basic inputs for determining both inflow into dam storage and crop irrigation requirements (i.e., dam outflow). When this is done, some of the combinations would possibly result in 'running out of water' in most or all of these 20 years, while other combinations would result in nonuse, i.e., wastage of most of the stored water most of the time. The optimal dimension of this dambuilding/resourcegenerating activity at, e.g., whatever level of reliability the analyst chooses to specify, would be determined by simulation.
Onstream water pumping. Here the resourcegenerating activity consists of installing a pump on a stream which has a random seasonal flow. The problem is to select the optimal size of pump. No dam is involved; water is pumped directly from stream to field. Necessary pump size will obviously vary inversely with the time (days) that the stream is expected to flow. Optimal pump size can most conveniently be found by simulating the operation of a range of pumps of various capacities and cost in relation to the water requirements of a range of different crops and/or crop areas.
FIGURE 11.14  Flowchart for Monte Carlo Simulation Model of Alternative Ricebased Farm Systems
Field and channel design for spateflood irrigation. In some dry regions where it is not uncommon for the total annual rainfall to occur in the form of one or a few intense storms of short duration, e.g., as in parts of Yemen, much irrigation is still done by controlled spate or wildflood methods. Runoff rushing down the wadi is diverted into the impounded fields and a farmer's annual grain crop might be grown entirely on the soil moisture from the spateflood from a single storm. There are three main technical planning problems on these farms. These are to determine (1) the optimal size/capacity of the waditofield delivery channel, (2) the size of the individual fields to be irrigated, and (3) the height/size of the polder embankments. If the fields are too small or the embankments too high, the excessive depth of impounded spate water will delay planting. If they are too large, this will result in insufficient soil moisture on which to grow the crop. Where the necessary rainfall and/or streamflow data are available, they would permit simulation of wildflood systems to determine these technical design parameters.
By their longterm nature, farm investment decisions are risky. This is true whether the decision relates to the whole farm (e.g., buying or selling a farm) or to some particular aspect of the farm's service matrix or inventory of capital items (e.g., investing in draught oxen), the outcomes of such decisions consist of a risky sequence of net returns over the period of the investment. These net returns may be in financial terms or in nonfinancial economic terms (i.e., in kind) such as, e.g., days of draught power, which may or may not be converted to financial terms on the basis of opportunity cost.
As noted previously, except on a casestudy or representativefarm basis, it is not to be expected that formal approaches to risky longterm decisions will ever be worthwhile for Type 1 (i.e., subsistence) or Type 2 (i.e., semisubsistence or partcommercial) farms. Formal approaches, as outlined in Sections 11.8 to 15, may be pertinent to larger farms and particularly to estates. Indeed, the archetypal longterm investment decision is that faced by an estate manager in deciding whether or not to replace an existing perennial tree crop with a new crop. Such decision problems are usually framed in the context of having payoffs constituting a stream of annual financial net returns which are converted to a PV by application of a discount rate which is assumed to be invariant over the life of the investment. A riskfree example is provided by the cardamom investment of Section 10.12 and a risky example by the banana investment of Section 11.15.2.
The time and riskoriented planning procedures of Chapter 10 and this chapter can undoubtedly provide some guidance to longterm farm investment decisions. However, these procedures ignore a number of problems (Anderson, Dillon and Hardaker 1977, Ch. 8; Hardaker, Huirne and Anderson 1997, Ch. 10; Robison and Barry 1996, Chs 11 and 23). First, even though it is subjective, the relevant discount rate is unlikely to be invariant over time. It is most likely to be uncertain and to fluctuate year by year into the future as it is influenced by changes in the market rate of interest and by inflation. Stochastic simulation models could incorporate such uncertainty. Second, an essential element of risky longterm investments is that their payoffs consist of not just a single probability distribution of outcomes but of a series of (perhaps serially correlated) probability distributions of outcomes extending (e.g., year by year) over the life of the investment. In comparing alternative investments, choice thus involves not choice between single probability distributions but between streams of probability distributions. Such choice is obviously a more difficult task. Usually it is assumed away by taking a sequence of particular payoffs and discounting this stream to a PV. Repetition of this process using Monte Carlo sampling can then be used to generate a probability distribution of PVs (as in the banana example of Section 11.15.2). This raises a third problem. The PV of a stream of net returns does not give any information about the sequential pattern of the stream. Two streams of net returns may have the same PV but markedly different sequential patterns, e.g., the following two net return streams:
Year: 
1 
2 
3 
4 
5 
Stream 1 ($): 
0 
0 
0 
0 
20 000 
Stream 2 ($): 
11 800 
0 
0 
0 
1 000 
have the same PV of $12 418 at a discount rate of ten per cent. Clearly, depending on their circumstances and the degree of imperfection in the capital market, different farmers would have different preferences between these two streams. PV is thus no sure basis for risky choice. It should be used with appropriate caution.
To conclude, it must be said that, while a variety of formal approaches to handling risky farm decisions have been developed, they are no panacea. Often their cost in terms of needed information and analysis will not be worthwhile. This is particularly so for small farmers. And when worthwhile, as may be the case for large commercial farms and estates, they can only provide guidance rather than sure direction. In sum, given the endemic nature of risk in farming and the complexity of farm decisions, a farm manager's risky choices must inevitably rely in part, or even in major part, on his or her intuition and subjective judgement.
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