Flat-earth economics: The far-reaching consequences of flat payoff functions in economic decision making

David J. Pannell

CRC for Plant-Based Management of Dryland Salinity and School of Agricultural and Resource Economics, University of Western Australia

Abstract

Economists tend to emphasise the optimum. For example, they ask questions such as, at what input level will expected profit or utility be maximised? However, the same empirical models used to identify the optimum also often tell us that even quite large deviations from optimal decisions may make little absolute difference to the expected payoff. This has far-reaching implications that are under-recognised. This paper is a review of some of the implications related to agriculture. Key conclusions flowing directly from flat payoff functions include: (a) decision makers often have a wide margin for error in their production planning decisions, and flexibility to pursue factors not considered in the calculation of payoffs; (b) optimising techniques are sometimes of limited practical relevance for decision support; (c) the value of information used to refine management decisions is often lower than might be expected; (d) there is a decreasing marginal value of precision in farming decisions, so the use of "precision farming" technologies to adjust production input levels is often of lower value than might be expected; and (e) representation of risk aversion in models used for decision support is often of low importance. A key question that modellers should ask themselves about the information they generate for decision support is, what difference will it make to payoffs?

Key words: economics of information, precision agriculture, economics of research, sustainability indicators, risk

Introduction

Within the neighbourhood of any economically "optimal" management system, there is a set of alternative management systems that are only slightly less attractive than the optimum. Often this set is large. For example, in the 1980s my colleagues and I modelled dryland crop-livestock farms in Western Australia to estimate the optimal percentage of land to allocate to crop production (Morrison et al., 1986; Kingwell and Pannell, 1987). We often found that within a wide range of cropping percentages the farm profit was within 10 percent of the maximum. Figure 1 shows an example where this is true for cropping percentage between 51 and 92 percent.

Figure 1. A typical result for Western Australian farms in the 1980s, showing a wide plateau with near-optimal profits (source: MIDAS model version 91-4, Pannell and Bathgate, 1991).

 

Models used to evaluate economically optimal levels of production inputs at the field scale also typically find a wide profit plateau. Examples in agriculture include models of production response to fertilisers, herbicides, insecticides and soil ameliorants. Figure 2 shows profit as a function of herbicide dose rate for the empirical model estimated by Pannell (1990). Although a dose of 0.26 kg ha-1 gives the maximum profit, any dose between 0.15 and 0.44 would yield profits within five percent of the maximum. In this sense, the margin for error is –40% to +70% around the optimum.

Figure 2. Profit from wheat production as a function of herbicide dose

 

The width and flatness of the profit plateau varies, but the presence of a profit plateau is almost universal in economic production models with continuous decision variables. This paper reviews and discusses the implications of flat payoff functions for decision makers and decision analysts, with a focus on agriculture.

The existence of flat payoff functions in agriculture was well recognised in the past, at least among production economists. Anderson (1975) cites a wide range of literature from the 1950s to the 1970s in which the issue was acknowledged and discussed, commencing with Hutton and Thorne (1955), Heady and Pesek (1955) and Hutton (1955). However, over time the issue has largely disappeared from the economics agenda, probably reflecting changes in the interests and skill base of agricultural economists.

Among non-economist agriculturalists, is appears that the issue was only ever recognised by a small minority. Jardine (1975b) noted that on presenting information to agronomists about flat profit curves for fertlizers, he "observed such reactions as complete disbelief, blank incomprehension, incipient terror, and others less readily categorized" (p. 200). I suspect that among many agronomists, the responses would be similar today.

Anderson’s and Jardine’s contributions were part of a spirited debate among researchers, prompted by Jardine (1975a) and including responses by White (1975), Godden and Helyar (1975) and Brown (1975). So far as I can tell, no similar exchange has graced the pages of any agricultural journal since, and even at that time some of the contributors to the debate had missed the point.

Despite the current neglect of them, flat payoff curves remain as frequent and as important as ever. Their implications are numerous and diverse and touch on some prominent modern trends in agricultural systems.

The next section discusses the origins and causes of flat payoff functions in economic models, particularly for agriculture. Then a wide range of implications and consequences of flat payoff functions are presented and discussed.

Causes of flat payoff functions

In the case of decision making about production inputs, the profit plateau arises from the usual shape of the production function. In agriculture, this generally reflects the biological behaviour of the production system. Figure 3 illustrates the absolute benefits and costs of wheat production as herbicide dose is varied. It uses the same production function as underlies Figure 2; the profit function of Figure 2 is the difference between the benefits and costs depicted in Figure 3. The benefit curve in Figure 3 directly reflects the way that crop yield responds (mainly due to decreasing weed competition) as herbicide dose is increased. It is obvious that whenever a production function takes this classic shape, as assumed in every microeconomics text, the smooth and concave-down shape of the curve in the region of the optimum will guarantee that there is some region where profits are very close to their maximum. The width of this region will depend on the biological and technical forces that forge the actual production function.

Figure 3. Benefits and costs of wheat production, treating herbicide dose as the only variable input.

 

Anderson noted that the issue does not only affect input response functions; that it is "a much more general phonomenon which pervades all optimisation processes and models" (p. 195), giving as examples whole-farm economic models and cost functions.

Where the decision is a choice among a portfolio of production options (e.g. Figure 1), the existence of a profit plateau is less assured than for input choices, but it is still common. A plateau is more likely if there is an internal solution to the portfolio problem (a mixture of production activities) rather than a corner solution (focus on a single option). A number of factors contribute to diversification of production activities in agriculture:

(a) Non-uniformity of resource quality (e.g. Morrison et al., 1986; Kingwell et al., 1992).

(b) Resource constraints, such as on machinery capacity (Pannell et al., 2000).

(c) Complementarity between enterprises, such as nitrogen fixation by legumes grown in rotation with cereal crops (Pannell, 1987).

(d) Risk aversion (Samuelson, 1967).

Whenever these factors combine to result in a diversified farm plan being optimal, there will exist a sizable set of near-optimal plans, as illustrated in Figure 1.

There are implications here for the degree of detail used by whole-farm modellers. Simpler farm models sometimes neglect to represent some of these factors, with the result that the existence of a flat region of the payoff function is not recognised. Often the failure of simple mathematical programming models to identify an internal solution is attributed to their non-representation of risk aversion, although the other factors listed above can be at least as important (Pannell et al., 2000).

Consequences and implications of flat payoff functions

Margin for error

A flat payoff function provides some comfort for the decision maker, in that there is a margin for error in the decision. A failure to optimise the decision will not be costly unless the decision made departs substantially from the optimum. In agriculture, farmers often do not take great care and detailed effort over their production decisions. Many farmers do not employ sophisticated analytical methods to support their decisions, and indeed many make production decisions on a somewhat intuitive basis. Where payoff curves are flat, this is probably a sensible prioritisation of effort, particularly if it allows farmers to allocate additional time to other aspects of management that are more likely to pay off (e.g. increasing their knowledge of production technologies and systems, investigating innovative practices).

Another consequence of a flat payoff function is that it empowers the decision maker to consider addition factors that are not reflected in the calculation of payoffs. For example, farmers often have personal preferences for, or interests in, some production options relative to others (e.g. a farmer may be comfortable with crop production and dislike dealing with animals). If the financial consequences of a choice between options are minimal, such personal preferences may become the decisive factor in the decision.

There are also implications for analysts developing decision support systems for farmers. The forgiving nature of many production decisions means that optimising techniques are of limited practical relevance. It is usually more helpful to the decision maker to identify the shape of the payoff function, and especially the range over which it is relatively flat, rather than emphasising a single optimal solution. This highlights the relevance of methods such as sensitivity analysis (Pannell, 1997). It does not imply that optimisation models should not be used, but has implications for the way that they should be used in decision support.

The value of monitoring sustainability indicators and other management indicators

During the 1990s there was a growing interest in the use of so-calls "sustainability indicators", in the hope that they would allow more "sustainable" management of agricultural lands (Pannell and Glenn, 2000). The issue is essentially one of providing additional information for use in management decisions, and is similar in many respects to the monitoring of standard economic variables, such as prices, interest rates, exchange rates, and debt.

One way or another, information costs money (the costs of experimenting, searching, communicating, storing, analysing, or buying information). An important question is whether the benefits of acquiring information exceed its cost. Flat payoff functions influence the result.

Before we consider that influence, we need some background on the estimation of information value. A decision theory framework (Anderson et al., 1977) is very helpful in understanding and estimating the value of information used to make management decisions. Here is a brief outline of the thought process involved in estimating the value of information in a farming context.

Suppose that a farmer has a decision to make, and the optimal decision depends on a number of variables, which I will call indicators. In general, the farmer will have some preconceptions about the indicators. Without making observations of the indicators, their values are not known with certainty, but the farmer has subjective views about the ranges within which they are likely to fall, and the likelihoods that they will take different values within those ranges. These preconceptions could be based on past experience, on observations of other farms or perhaps on external advice.

It would be possible for the farmer to go ahead and make a best-bet decision based solely on his or her preconceptions. Alternatively, the farmer could make observations of relevant indicators before making the decision. With the extra information from these observations, an improved decision may be possible. The extent of improvement depends on issues such as:

and so on. Considering issues such as these, it is possible to make an assessment of the likely benefits from observing the indicators. The expected value of the information obtained by observing the indicators is the difference between the expected benefits and the costs of collecting the information.

Pannell and Glenn (2000) provided a numerical example that will help us to explore the issue. A farmer must choose the area of land that should be planted to trees. Land not devoted to trees is used for production of wheat. Assumptions and parameters of the model are presented by Pannell and Glenn (2000). The payoff curves for two discount rates are shown in Figures 4 and 5.

Figure 4. Payoff for decision on area of trees (discount rate 10%)

 

Pannell and Glenn (2000) found that the value of collecting additional information to support the decision was low; zero in some cases. There are different reasons for the low values in the two cases, 10 percent and 15 percent discount rates. In the 10 percent case, the payoff function is very flat (Figure 4). As a result, even if the farmer chooses an area of trees that differs substantially from the true optimum, the loss of profit relative to the optimum is low. If refining decisions does not make much difference to the payoff, the value of monitoring the indicators is low.

On the other hand, with a discount rate of 15 percent, the optimal choice (to grow no trees) is so clear that there is no information that would alter it. Since the additional information cannot alter the decision, it cannot change the payoff, and so has zero value.

Agricultural research or extension organisation often encourage farmers to monitor various variables on their farms, but the argument presented above presents such organisations with a serious conundrum. Ideally, they would like the payoff to farmers from monitoring their proposed indicators to be high. However, if the treatment is either highly advantageous or clearly uneconomic, the best decision is obvious and the value of any further information will be low. Conversely, if the payoffs from different decision options are similar enough so that the optimal decision is unclear and can be clarified by further information, then the overall payoff curve is likely to be flat, so that again the information is of limited value.

Figure 5. Payoff for decision on area of trees (discount rate 15%)

 

The value of precision farming

"In pursuing … optimal levels of decision variables, precision is pretence and great accuracy is absurdity," (Anderson 1975, p.195).

A second information-related example relates to so-called "precision farming", an innovative approach to managing agricultural inputs. This term encompasses a range of technologies and approaches, but all involve the use of information to adjust levels of an agricultural input to suit conditions in different locations in a field. This spatial variability of inputs recognises that physical conditions also vary spatially, so that the optimal input level is higher or lower in different locations. Some of the technologies involved are expensive, so accurately estimating the benefits derived from precision farming is important to farmers. The issues are fundamentally the same as those described above for monitoring of an indicator for improved decision making by farmers, but they operate on a micro-scale and with automated decisions. If the precision farming technology results in low to moderate changes in input levels at a given location within the field, there is a high probability that the improvement in profit will be very low due to a flat payoff function.

Bennett and Pannell (1998) found very low benefits from use of a precision agriculture system for herbicides. Flat payoff functions contributed to that result, although corner solutions (or near-corner solutions) were the main causal factor.

In general, flat payoff functions mean that there are diminishing marginal returns to precision in decisions about agricultural inputs. O’Connell et al. (1999) quantified the extent of this diminution for a particular example: application of lime to treat soil acidity. Their example considered three levels of information use and precision in the decision.

1. Very low information use/precision: The same rate of lime would be applied in all situations.

2. Low information use/precision: Generalised recommendations were made for each soil type and each rotation in each of three regions, based on typical bio-physical conditions for those situations.

3. Moderate information use/ precision: Bio-physical information was collected on a field by field basis to refine the recommended rate to best suit that particular situation.

One could envisage a fourth strategy involving high precision, in which the rate would be varied within each field, but this was not examined in the study.

Table 1 shows the gross values of increasing precision from very low to low, and from low to moderate. It reveals a marked reduction in the marginal value of precision in this decision. One would anticipate that the additional value of even greater precision in this decision (as in a "precision farming" approach) would be close to zero. The reason for this diminishing marginal value is the flat payoff function for this problem. (An example for one of the scenarios of the analysis is shown in Figure 6). Once the precision of a decision is high enough to ensure a high probability of targeting a rate within the payoff plateau, further precision has very limited scope to improve the payoff.

 

Table 1. Incremental benefits (A$/ha/year) of increasing the information intensity of decisions about lime application to treat soil acidity (Source: O’Connell et al., 1999)

Zone Soil type Change from very low to low precision Change from low to moderate precision

Low rainfall

Deep sand

14

4

Clay

8

2

Medium rainfall

Deep sand

35

3

Clay

19

2

High rainfall

Deep sand

7

3

Clay

21

0

Source: O’Connell et al. (1999).

Figure 6. A typical payoff function for lime application underlying the result in Table 1. (Source: O’Connell et al., 1999)

 

The value of research and extension

Flat payoff curves affect the likely values of research and development (R&D) of different types: R&D to develop new technologies versus R&D to generate information to improve decision making about existing technologies (Pannell 1999). In conducting evaluations of the benefits of R&D (e.g. Alston et al. 1995), it is not uncommon for analysts to fail to distinguish between these two categories. However, because of flat payoff functions, the distinction can be crucial for accurate estimation of benefits.

For the second category of R&D (improved decision making about existing technologies) it will often be the case that R&D only allows refinement of management decisions within a payoff plateau. In this case, the extent of improvement in payoff that results from the R&D will be low. In my judgment, it is common for analysts to overstate the benefits of this category of R&D, sometimes substantially so. Sometimes analysts confuse information that a technology is more profitable than previously believed with an actual improvement in its profitability. They may not recognise that a change in management in response to the new information still involves movement along a flat payoff function.

To illustrate, consider Figure 7. It shows the payoff to a hypothetical (but typical) agricultural input. Without research, the perceived payoff to different input levels follows the lower curve. Suppose that research provides information that the true payoff curve is substantially higher. What is the value of this information? Without research, the input level that appears to be optimal is I1 and the anticipated payoff from this input level is P1,1. After the research, the optimal input level is revealed to be I2 and the anticipated payoff from this input level is P2,2. However, note that the research does not actually shift the payoff function, it only provides improved information about it. If we assume that the perceived payoff function with research is actually true, then application of the original input level I1 would have resulted in payoff P1,2. The improved payoff resulting from the research is the short vertical distance between P1,2 and P2,2.

Figure 7. The value of information from information that a payoff function is higher than expected is P2,2 - P1,2.not P2,2 - P1,1.

 

On the other hand, the outcome of the first category of research (to develop new technologies) would be more likely to be an actual rise in the payoff function (e.g. through breeding of higher yielding crop varieties). If the shift in the payoff function illustrated in Figure 7 was an actual shift resulting from R&D, the value of the R&D to this decision maker would be the full difference P2,2 - P1,1. The benefits of this type of R&D are not compromised by the presence of any flat payoff function. From this it can be seen that, other things being equal, there are good reasons to expect that successful R&D intended to improve management decisions about existing technologies will pay off less than successful R&D to develop new, higher performing technologies.

For similar reasons, payoffs from agricultural scientific communication (extension or technology transfer) that provides information useful for management decisions about existing farming technologies will also struggle to overcome the influence of flat payoff functions. Unless the information suggests management practices that are substantially different to those in current use, increases in payoffs are unlikely to be large. This suggests the need for extension to target issues where the technologies are new and unfamiliar to farmers, or where for some reason farmers have developed clear and important misperceptions about the technologies.

Risk

Risk aversion on the part of a decision maker generally only makes a modest difference to optimal decisions relative to a risk-neutral decision maker. As we have seen, when payoff functions include wide flat regions, modest differences to decisions often translate in very small benefits to the decision maker. From the point of view of a decision analyst, this can mean that inclusion of risk aversion in models intended to be used for decision support is of low priority (Pannell et al., 2000).

To understand this conclusion, it is necessary to appreciate that if a graph of expected profit versus the decision variable includes a flat region near the optimum, then so too will a graph of "certainty equivalent" (CE) (i.e. expected profit minus a risk premium), although it will probably be slightly shifted left or right. Now, suppose an advisor ignores the fact that a farmer is risk averse and uses a risk-free model as the basis for advice to the farmer. In many cases, such an oversight will matter little. If the payoff function (in this case representing CE rather than profit) is flat, small to moderate errors in decisions make little difference to the payoff. Table 2 shows how little difference it makes, for a decision based on the herbicide dose model cited earlier. The results are for levels of relative risk aversion up to 3.2, which is considered high.

 

Table 2. Cost to farmers in Western Australia in terms of lost potential certainty equivalent from implementing the herbicide dosage from a model with a risk-neutral objective function

Farmer’s relative risk aversion

Cost of using risk-neutral model

(%)

0.0

0.0

0.8

0.038

1.6

0.18

2.4

0.46

3.2

0.95

Source: Pannell et al. (2000).

 

In relation to these results, Pannell et al. (2000) observed that, "The primary reason for the low impact of risk aversion lies in the unresponsiveness of certainty equivalent to changes in farm management within the region of the optimum" (p. 72). They noted that, "Because of this flatness, consideration of complexities such as risk aversion, which only change the optimal strategy by moderate amounts, does not greatly affect farmer welfare. Thus the argument is not that risk aversion does not affect the farmer’s optimal plan, but that the impact of the changes on farmer welfare is small." (p. 72).

As a final example, Figure 8 shows the consequences of mis-specifying a producer's risk aversion coefficient in a representative farm model in north-west Syria (Pannell and Nordblom, 1998). Each line in Figure 8 represents the certainty equivalent value of income for a hypothetical Syrian farmer with a utility function characterised by constant relative risk aversion. The different lines represent farmers with different levels of relative risk aversion (RRA). Moving from left to right along a line corresponds to different combinations of enterprises and practices identified as optimal by models of increasing risk aversion. On each line, at only one point does the risk aversion coefficient of the model correspond to that of the farmer and, at this point (which is marked), the certainty equivalent value to that farmer of the plan recommended by the model is maximised. All other models recommend a plan of lower value to that farmer.

The most striking aspect of the figure is the flatness of the lines, reflecting that the certainty equivalent value of a farm model’s recommended plan is insensitive to the degree of risk aversion used to derive the plan. If a risk-averse farmer implemented a plan that would be optimal for a risk-neutral farmer, the payoff foregone by the farmer would be small. Even at the highest level of risk aversion (RRA = 3.2), the cost in certainty equivalence of ignoring risk aversion is less than two percent. This is despite the fact that climatic risks faced by Syrian farmers are among the highest in the world (Nguyen, 1989).

Figure 8. Certainty equivalent values of income for farmers in north-west Syria based on implementation of whole-farm plans from models representing various degrees of risk aversion. (Source: Pannell et al., 2000)

 

Conclusion

The implications of flat payoff functions in economic models have not received the attention that they deserve. Economists in universities teach their students how to identify optimal solutions to various types of decision problems, and applied economists and operations researchers advise decision makers about optimal management strategies, but in each case they sometimes fail to mention that payoff curves are generally flat near their maxima. Since this flatness is of far greater practical significance than the identification of the optimum, such a failure warrants criticism. Perhaps decision analysts should take greater efforts to address a key question about the information they generate: what difference will it make to payoffs?

The insights presented here have implication for the type of advice that decision analysts addressing economic problems should give to decision makers, and for the way that decision problems should be analysed. The implications of flat payoff functions include that:

(a) decision makers often have a wide margin for error in their production planning decisions, and flexibility to pursue factors not considered in the calculation of payoffs;

(b) for many types of problems, optimising techniques are of limited practical relevance for decision support;

(c) the value of information used to refine management decisions is often lower than might be expected;

(d) there is a decreasing marginal value of precision in farming decisions. The use of "precision farming" technologies to adjust production input levels is often of lower value than might be expected;

(e) representation of risk aversion in models used for decision support is often of low importance.

Overall, these insights are important to decision analysts in prioritising the approach they take to economic modeling in various contexts, particularly for decision support. There may be unmet opportunities for decision analysis to improve the welfare of decision makers through greater appreciation of these insights.

References

Anderson, J. R. (1975) One more or less cheer for optimality. Journal of the Australian Institute of Agricultural Science, 41: 195-197.

Anderson, J.R., Dillon, J.L. and Hardaker, J.B. (1977) Agricultural Decision Analysis. Ames: Iowa State University Press.

Bennett, A.L. and Pannell, D.J. (1998). Economic evaluation of a weed-activated sprayer for herbicide application to patchy weed populations. Australian Journal of Agricultural and Resource Economics 42(4): 389-408.

Brown, K.J. (1975). Another opinion, Journal of the Australian Institute of Agricultural Science, 41: 198.

Godden, D.P., and Helyar, K.R. (1975). Optimality, Hurrah! Journal of the Australian Institute of Agricultural Science, 41: 197-198.

Heady, E.O. and Pesek, J. (1955). Hutton and Thorn on isoclines: a reply, Journal of Farm Economics 37: 363.

Hutton, R.F. (1955). Further comment on the Heady-Pesek fertilizer production function, Journal of Farm Economics 37: 566.

Hutton, R.F. and Thorne, D.W. (1955). Review notes on the Heady-Pesek fertilizer production surface, Journal of Farm Economics 37: 117.

Jardine, R. (1975a). Two cheers for optimality. Journal of the Australian Institute of Agricultural Science, 41: 30-34.

Jardine, R. (1975b). Jardine’s rejoinder. Journal of the Australian Institute of Agricultural Science, 41: 199-201.

Kingwell, R.S. and Pannell, D.J. (Eds) (1987). MIDAS, A Bioeconomic Model of a Dryland Farm System, Pudoc, Wageningen, 207pp.

Kingwell, R.S., Morrison, D.A. and Bathgate, A., 1992. The effect of climatic risk on dryland farm management, Agricultural Systems, 39, 153-175.

Morrison, D.A., Kingwell, R.S., Pannell, D.J. and Ewing M.A. (1986). A mathematical programming model of a crop-livestock farm system, Agricultural Systems 20(4): 243-268.

Nguyen, H., (1989). Agricultural planning policy and variability in Syrian cereal production. In: Anderson, J.R. and Hazell, P.B.R. (Eds.), Variability in Grain Yields: Implications for Agricultural Research and Policy in Developing Countries, pp. 78-90, Johns Hopkins University Press, Baltimore.

O’Connell, M., Bathgate, A.D. and Glenn, N.A. (1999). The value of information from research to enhance testing or monitoring of soil acidity in Western Australia, SEA Working Paper 99/06, Agricultural and Resource Economics, University of Western Australia.

Pannell, D.J., 1987. Crop-livestock interactions and rotation selection. In: Kingwell, R.S. and Pannell, D.J. (Eds.), MIDAS: A Bioeconomic Model of a Dryland Farm System, pp. 64-73. Pudoc, Wageningen.

Pannell, D.J. (1990). An economic response model of herbicide application for weed control in crops, Australian Journal of Agricultural Economics 34(3): 223-241.

Pannell, D.J. (1997). Sensitivity analysis of normative economic models: Theoretical framework and practical strategies. Agricultural Economics 16: 139-152.

Pannell, D.J. (1999). On the estimation of on-farm benefits of agricultural research. Agricultural Systems 61(2): 123-134.

Pannell, D.J. and Bathgate, A. (1991). MIDAS, Model of an Integrated Dryland Agricultural System, Manual and Documentation for the Eastern Wheatbelt Model Version EWM91-4, Miscellaneous Publication 28/91, Department of Agriculture, Perth, Western Australian, 162 pp.

Pannell D.J. and Glenn N.A. (2000). A framework for economic evaluation and selection of sustainability indicators in agriculture, Ecological Economics 33(1): 135-149.

Pannell, D.J. and Nordblom, T.L. (1998). Impact of risk aversion on whole-farm management in Syria. Australian Journal of Agricultural and Resource Economics 42(3): 227-247.

Pannell, D.J., Malcolm, L.R. and Kingwell, R.S. (2000). Are we risking too much? Perspectives on risk in farm modelling. Agricultural Economics 23(1): 69-78.

Samuelson, P.A., 1967. A general proof that diversification pays, Journal of Financial Quantitative Analysis, 2, 1-13.

White, D.H. (1975). The search for an optimal stocking rate: continuing the saga, Journal of the Australian Institute of Agricultural Science, 41: 192-194.

Citation: Pannell, D.J. (2004). Flat-earth economics: The far-reaching consequences of flat payoff functions in economic decision making. Unpublished paper presented at 48th Annual Conference of the Australian Agricultural and Resource Economics Society, Melbourne, 11-13 February 2004. http://www.general.uwa.edu.au/u/dpannell/dp0402.htm 
Revised version published as:
Pannell, D.J. (2006). Flat-earth economics: The far-reaching consequences of flat payoff functions in economic decision making, Review of Agricultural Economics 28(4): 553-566. 

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