191

UNCERTAINTY, IRREVERSIBILITY AND THE LOSS OF AGRICULTURAL LAND

Ian Hodge* University of Queenslandt

In the past, evahations of the Iransfer of agricultural land to other uses have generally failed to incorporate the issues of uncertainty and irreversibility into the analysis. This can lead to an overestimation of development values. The paper commences with a brief review of the approaches which have been adopted in the assessment of the loss of agricultural land. Ir is then demonstrated that uncertainty relating to the rate of change of agricultural values through time requires the adoption of a more conservative decision criterion in the evaluation of irreversible land use changes than would otherwise be applied. This is followed by a numerical example and sensitivity analysis. The paper concludes with a discussion of these results.

Introduction The issue of the loss of land from agricultural uses is one which has provoked a lengthy and widespread debate (see e.g., Crosson, 1982a, Hall, 1980, Leslie and Johnston, 1982, for some recent contributions). In this paper, i t is argued that the application of cost-benefit principles to the question of the transfer of land between uses represents the approach which is likely to make the greatest contribution to rational decision-making at the micro-economic level. I t is then demonstrated that, in this approach, under conditions where the transfer is irreversible and when there is uncertainty about the future, a change in land use cannot be justified by showing simply that t h e expected present value of the non-agricultural land use exceeds the expected present value of the agricultural use. Under these conditions, which a re likely to apply in practice, i t is necessary to demonstrate that the value of the non-agricultural use should exceed the value of the agricultural use to a predetermined extent. The size of this depends primarily upon the degree of uncertainty about the future and upon the discount rate. The significance of this effect is demonstrated under a range of assumptions.

Assessment of the Loss of Agricultural Land A number of approaches have been taken in studying the question of the loss of farm land. At their most fundamental, have been attempts to measure the extent and the quality of the-land which has already been transferred to urban

* Most of the research for this paper was undertaken while the author was engaged in a Special Studies Programme in the Department of Agricultural Economics at the University of Idaho. Thanks are due for the provision of facilities rhere and in particular to Professor Joel Hamilton for his assistance and encouragement. I am also grateful to members of the Department of Agricultural Economics at [he University of Newcastle upon Tyne for their heluful comments.

7 Current address: Department of Land Economy, Universiry of Cambridge. 19 Silver Street. Cambridge CB3 9EP.

192 IAN HODGE

use. Despite considerable research effort these have failed to produce agreement on either the physical changes or on their implications (e.g., Rogers, 1978; Best, 1981; Brewer and Boxley, 1981).

A second, more analytical, approach has been to seek to project the demand and supply of land on a national basis over some future time period. In the UK, these studies have been based on the idea of a ‘land use budget’ (Edwards and Wibberley, 1971; Centre for Agricultural Strategy, 1976). These studies have commonly been framed so as to identify the area of agricultural land required to achieve a specified level of national self-sufficiency at some future date. American studies have related to the objective of meeting rising export demands for food and fibre (Crosson, 1982b). However, the conclusions to be drawn from these models are far from clear (Champion, 1977; Wise and Fell, 1978). Unless the given level of self-sufficiency in food production or the fulfilment of the specified level of export demand can be taken as overriding objectives, irrespective of the costs involved, the normative implications of such research are rather limited. The value of this approach lies in providing information on the trade-offs between alternative objectives, such as relating to food security, food prices and trading arrangements.

A third approach has sought to apply the principles of cost-benefit analysis to these issues. This first attempted to account for differences in housing construction costs on different qualities of land (Ward, 1957; Wibberley, 1959) and subsequently concentrated on the evaluation of the losses associated with the withdrawal of agricultural land from production (Boddington and Wibberley, 1970; Boddington, 1973). While the evaluation of many of the factors involved presents severe difficulties, an attempt to compare the costs and benefits associated with land use changes represents the only approach to questions of the transfer of individual blocks of land from agricultural use. The first two approaches provide background data for this decision.

The Value of Land Retained in Agriculture Primarily, agricultural land provides services as a consequence of its role as one input in the agricultural production process. However, particularly in the debate over the retention of farm land, other benefits may be ascribed to it. Hite and Dillman (1981) have reviewed the reasons behind moves to protect agricultural land, beyond the question of maintaining adequate food supplies at a national level. The demands for the protection of agricultural land appear to stem from four general areas of concern. First, there is concern about future food production potential to meet demands for food at a global level. Second, concern is expressed about the security of food supplies at a national or regional level. Third, there is concern about the physical consequences of uncontrolled urban development, either in terms of the external costs associated with a reduced agricultural area, or in terms of the impacts of urban development, such as increased costs of providing services to a low density urban area. Finally, there may be concern over the multiplier effects on the regional economy.

In that agricultural benefits are not necessarily the only benefits to be gained from farm land, then their evaluation alone cannot provide a full measure of the costs associated with land conversion. Nevertheless, much of the general concern voiced about possible farm land shortages does concentrate on the role of land in the production of food, fibre and, more recently, energy (e.g., Bowman, et d. 1978; Little 1979; Brown, 1980). A major cause of this concern lies in the fact that much of this loss is not reversible. This issue has not been

UNCERTAINTY, lRREVERSl8 lL lTY A N D THE LOSS OF AGRICULTURAL LAND 193

accounted for analytically in the evaluation of agricultural land, although Peters (1970) has commented that market forces should not necessarily hold sway in determining land allocations, ‘for this might lead to a position in which higher grade land was taken out of production in an irreversible manner, without regard to longer term considerations’.

The Problem of Irreversibility and Uncertainty A variety of the changes which can occur in land use are effectively irreversible. Henry (1974) considers a decision to be irreversible if it ‘significantly reduces for a long time the variety of choices that would be possible in the future’. Clearly, in practice, irreversibility is a question of degree; some decisions are more reversible than others. At an extreme, the destruction of an ancient monument must be totally irreversible in the sense that the culture which constructed it cannot be recreated. However, in many circumstances, the time required for the decision to be reversed, such as in the regrowth of a hardwood forest, or the cost, such as in the return of urban land to agriculture, make the decision effectively irreversible from the point of view of current decision-makers.

Changes involving the loss of land from agriculture can be of this latter category (e.g., McInerney, 1976). There are several such types of change. The most obvious is the process of urbanisation where the flow of agricultural production services is replaced by a flow of urban land services. This change effectively removes the option of regaining the agricultural services from that land due to the high costs of reclamation. Other types of change also exhibit this characteristic. The loss of land through soil erosion must be regarded as irreversible due to the extremely slow rate of soil formation. Similarly, the degradation of land in open cast mining and the inundation of land for water storage is likely to be irreversible. While the discussion in this paper concentrates on the loss of land through urbanisation, the same principles apply to these other examples.

The impact of irreversibility on decision-making has been analysed by Arrow and Fisher (1974), Henry (1974) and Miller (1981). They demonstrate the need for caution in making irreversible decisions where there is uncertainty about the future. In particular, they show that, for a risk-neutral decision- maker under uncertainty, the expected value of benefits from an irreversible decision is less than the value of benefits under certainty.

The effect arises as a result of foreclosing possibly valuable future options as a consequence of some earlier irreversible decision. The value of the increased information which becomes available through time, can only be exploited i f the decision to develop is deferred. The implication is that where investment decisions are based solely on expected values, there will tend to be an overinvestment. Thus, in the case of the land conversion process, the benefits of development will tend to be over-estimated unless the uncertainty involved is taken into account. Arrow and Fisher refer to this reduction in net benefits as a ‘quasi-option value’. As Henry demonstrates, this irreversibility effect decreases with an increasing discount rate and increases with increasing uncertainty. Some modification of conventional decision criteria is therefore appropriate in order to avoid this loss.

Irreversibility and Changing Farm Land Use-Values The variability in the annual returns to land from any particular economic activity may be divided between the variation about the mean value due to a large variety of stochastic influences, and variation of the mean, due to more

194 IAN HODGE

steady and fundamental influences. For instance, short-run stochastic variations will result from such factors as weather, pests and political influences in world markets. Longer-run trends will be caused by changes in agricultural productivity and the influence of population size and income levels on the demand for agricultural products. In evaluating development choices in practice, it is quite likely that the analyst will seek to identify not only the expected mean value, but also the expected trend in this mean through time. He wiil, of course, be uncertain as to the actual magnitude of this trend. Where the consequences of the decision are irreversible, this uncertainty also leads to a need to apply a more conservative decision criterion.

Imagine that the annual value of development (0,) is known and is fixed through time ( t ) . The initial annual value of preservation (Po) is known but is subject to some constant annual rate of change (g) through time, which is unknown. In each year, the actual preservation value realised is equal to the initial value adjusted for growth (i.e., P, = Po (1 + g ) f ) . Thus, short-run stochastic influences are excluded from the analysis. These would in practice restrict knowledge of g.

First, assume that the decision-maker takes no account of the possibility of any change in the preservation value and adopts a simple benefit/cost ratio criterion (i.e., equal to 1). He will therefore choose to develop if Do > Po. But in reality he should only develop if

T c T

t = O

D, / (1 + r)' r = O L 1,

c Po(l + g ) ' / ( l + r)'

where r is the discount rate and T is the time horizon beginning at year zero. If an infinite time horizon is used and where r > g , this may be rewritten as

Po / [ l - (')I 1 + r

or, simplifying, as

This may be rearranged so that the project should be accepted where

Thus, if the actual decision is based on Do / Po, then the decision criterion (Q) should be set equal to ( r / r - g). The implications of different growth rates with alternative discount rates for the appropriate decision criterion are

UNCERTAINTY, IRREVERSIBILITY A N D T H E LOSS OF AGRICULTURAL LAND 195

shown in Table 1. Thus, for instance with a discount rate of five per cent and a rate of growth of preservation value of three per cent, the decision to develop should only be taken if Do / Po >= 2.5.

Table 1 Appropriate Decision Criteria (Q) ~

g r = 0.05 r = 0.1

+0.030 2.500 1.429 +0.020 1 ,667 1.250 + 0.010 1.250 1.111 + 0.005 1.111 1.053

0.000 1.000 1.000 - 0.005 0.909 0.952 - 0.010 0.833 0.909 -- 0.020 0.714 0.833 - 0.030 0.625 0.769

In reality, the problem is not that the possibility of changing values is ignored, but rather that growth rates are unknown. I f it is assumed that the distribution of growth rates is known, then it follows that the chance of any particular decision criterion being appropriate is also known. Further, the probability of using any decision criterion incorrectly is known too. If the loss from using the incorrect decision criterion can be calculated, then the decision criterion which produces the smallest loss would be the appropriate one.

Losses from Selecting Incorrect Growth Rates If Q is fully adjusted for any particular expected rate of growth (g), which in fact turns out to be g , then a loss could result from the selection of the incorrect option. The size of this loss is equal to the difference between what is achieved, given that a particular decision has been made, and what could have been achieved if the correct decision had been made. It depends upon Do, Po, 2 and g. It is possible that the difference between g and g would not have been sufficient to influence the decision and therefore would not have caused any loss. Where there is a large difference between Do and Po, a relatively small error in 2 is unlikely to influence the decision and so is unlikely to lead to a loss. As Do and Po approach equality, the importance of the accuracy of 2 increases. The maximum loss would occur if the decision was made when the expected present value of development was equal to the expected present value of preservation (i-e., where the decision-maker was indifferent between the two options based on g). Losses could result either where g > 2, so that the actual growth in preservation value exceeded the expected growth or where g < g where the preservation va:.iie less than expected.

(i) In the former case, the potential error would be to choose development when, in fact, preservation would have generated a higher present value. The loss

Losses where development is chosen

196 IAN HODGE

involved would be equal to the difference between what was actually achieved (i.e., the development option) and what could have been achieved (i.e., the preservation option with realised rate of growth). Thus,

The maximum loss will be made where development is chosen and the benefit from development is equal to the expected benefit from preservation, i.e.,

Thus, the maximum loss (ML) can be expressed solely in preservation value.

7- Po(l + 8)' PO(1 + g)' I - c

t = O ( 1 + r ) t = 0 (1 + M L = c

With an infinite time horizon,

(6)

terms of the

(7)

(ii) Losses where preservation is chosen In the latter case where g < 2, i.e., where preservation values grow less than expected, the loss will result from the choice of preservation when development would have been more valuable. In this case the loss will be given by

Once again, the maximum loss will be sustained where the decision-maker was indifferent between the development benefits and the expected preservation benefits, so that the maximum loss will be given by

The consequence of the irreversible nature of the development decision leads to an asymmetry in the size of the losses involved. In the first case, where development was chosen, the irreversible nature of this decision means that the losses must be sustained indefinitely. However, in the second case, where preservation was chosen, a decision can be made to develop at a subsequent date, so that the losses can be reduced when more information becomes available. There are two components of the losses involved. The first of these results simply from a lack of. information leading to a potential error in the initial decision. This applies equally to the choice of preservation when development should have been chosen and vice versa. The second applies only to the latter error and results from the irreversibility constraint. This component is thus a user cost as defined by McInerney (1976).

UNCERTAINTY, IRREVERSIBILITY AND THE LOSS OF AGRICULTURAL LAND

An Estimation of the Appropriate Decision Criterion in the Context of Irreversibility The consequences of this can be demonstrated through an example. Suppose that a decision is made in year 0 on the basis of the expected rate of change of preservation value. If, after a period of time the true value of this rate of change becomes known, then it would be possible to correct for the selection of preservation value. If , after a period of time the true value of this rate of change assumed that there is no cost associated with the correction itself. Thus, the maximum possible loss from choosing development in error would be given by equation (8), while the maximum loss from choosing preservation in error would be given by equation (lo), with T equal to the period of time taken to gain the information required for the correction of the decision.

The significance of this difference in maximum losses can be evaluated if the probability distribution of growth rates is known. Assume that this is as shown in Table 2.

197

g

+0.020 +0.010 +0.005

0.000 -0.005

-0.010 -0.020

Table 2 Probability Distribution of Growth Rates

probability of g

0.1 0. I 0.2 0 . 2 0 . 2 0.1 0. i

If the question of irreversibility is ignored, the expected rate of growth of preservation would be 0 and the appropriate decision criterion would be 1 , so that development would be undertaken where Do > Po. This would minimise the expected losses. However, if the decision is irreversible, it is possible to calculate the expected losses on the basis of the selection of alternative decision criteria.

For each expected rate of change there is an appropriate decision criterion, as shown in Table 1. I t can be assumed that growth will take place at any of the seven possible rates shown in Table 2. For each one, the expected maximum loss can be calculated in relation to the initial preservation value from

7 C PRjML;.

i = 1

PR; is the probability of each rate of change actually occurring and ML; is the maximum loss which can result from each combination of expected and actual rates of growth. Thus, the expected maximum loss can be calculated for each decision criterion in terms of the initial preservation value.

198 IAN HODGE

1

For instance, if the expected rate of growth of preservation value is 0.5 per cent and the discount rate is 10 per cent, then the appropriate decision criterion (Q) would be 1.053, as shown in Table 1. Assume also that it takes 20 years for the information necessary to correct any error to become available. The maximum loss can then be calculated, given this expected rate (g), for each actual rate (g). The results are shown in Table 3.

2 I 3 I 4

Table 3 Calculation of Expected Maximum Loss with Expected Growth of Preservation of 0 .5 Per Cent.

g

+0.020 +0.010 +0.005

0.000 -0.005 -0.010 -0.020

Probability Maximum (2 x 3) LOSS

0. I 2.171 0.2171 i?>% 0. I 0.643 0.0643 3 ... loss over T = Qo

0.2 0.000 0.0000 k!<i 0.2 0.328 0.0656

0.2 0.640 0.1280 0. I 0.936 0.0936 0. I 1.485 0.1485

. . loss over T = 20 1

B -

7 -

6 - Expected Maximum 5- Loss ( *Pol '-

3 -

Expected Maximum Loss = 0.7171

- - - - r = 5 per cent r -1Oper cent

\ \ \ \ \ \ \ \.

\ \ . __-. \ '.

..- __._ -- - -- 2 -

1 -

Column 3 shows the maximum loss, as a function of Po, experienced for each actual rate of growth. In the first two rows the potential error made would be to choose development when preservation should have been chosen and the loss is measured over infinity. In the bottom four rows, the potential error would be to choose preservation when development should have been chosen, so that the loss is measured over 20 years. The expected maximum loss is calculated by multiplying column two by column three and summing the results. This gives a total of 0.7 17 1. This exercise is repeated for each expected growth rate. The results, adopting discount rates of 5 and 10 per cent, are shown in Table 4 and Figure 1.

Figure 1 Expected Maximum Losses from Adopting Alternative Decision Criteria.

I I

1.05 1.25

I I I 1 I I I I I 0.6 0.8 1.0 1.2 1.L 1.6 1.8

OL, Q

UNCERTAINTY, IRREVERSIBILITY AND THE LOSS OF AGRICULTURAL LAND 199

I Discounr Rare = 10%

Table 4 Expected Maximum Losses from Adopting Alternative Decision Criteria

Discouni Rate = 5%

1.250 I . 384 1.111 0.865 1.053 0.717 1.000 0.752 0.952 0.941 0.909 1.263 0.833 1.958

I Decision Expected Maximum Decision Expected Maximum Criterion Loss Crilerion Loss

1.667 2.426 1.250 2.113 1.111 2.212 1.000 2.803 0.909 3.733 0.833 4.903 0.714 7.060

Discount Rare (per cent)

As can be seen, the expected maximum loss is minimised through the use of a decision criterion of 1.05 with a 10 per cent discount rate and by a criterion of 1.25 with a 5 per cent discount rate. Furthermore, the losses arising from overestimating the growth rate are lower than those resulting from underestimating the growth rate. This analysis is based on an attitude of indifference to risk. An attitude of risk aversion on the part of the decision- maker would encourage a more conservative, i.e., higher, decision criterion.

Time Before Correciion of Decision (Years)

10 20 30 50

Sensitivity .4nalysis The calculation of the decision criterion which minimises the expected maximum loss has been repeated under alternative assumptions. Discount rates of 3, 5, 10 and 15 per cent and periods of 10, 20, 30 and 50 years taken to obtain the information necessary to correct the decision, were tested. The decision criteria required to minimise the expected maximum losses are shown in Table 5.

3

5 10

15 1 .oo I .oo

3.00 3.00 3.00

1.67 1.25 1 . 1 1

1.11 1.05 1 .OO 1.03 1 .oo I .OO

I I I

Both of these parameters have a significant effect on the result. If a discount rate of 3 per cent is selected, a decision criterion of 3 is appropriate in all cases, except where 50 years is allowed to acquire sufficient information to correct the decision. On the other hand, a discount rate of 15 per cent gives a decision criterion of one or very close to one. Variation in the period of time before the decision can be corrected has a less dramatic effect on the result.

200 IAN HODGE

In general, a discount rate within the range of 5 to 10 per cent might be adopted. It is felt that a period of about 20 years could be required in order to acquire sufficient information so as to be able to correct a land transfer decision. In practice the decision-maker will not have full knowledge after 20 years. However, at that time knowledge of the values relevant to the initial year (0) decision will be substantially improved. Much land use planning concentrates on periods of about 20 years into the future. For instance, the year 2000 has been the target year for a number of land use studies in the 1970s and 1980s (Centre for Agricultural Strategy, 1976; National Agricultural Lands Study, 1981). Adoption of these parameters suggests that a decision criterion within the range of 1.05 to 1.25 could be applicable with this distribution of growth rates. In fact, the decision is more complicated than this. At the end of whatever time period is selected, there will always be the possibility of further information becoming available beyond that time. It could be possible to develop a more sophisticated, multiperiod model, where the probability of gaining information about the initial decision declines through successive time periods.

The wider is the distribution of growth rates the greater will be the impact of irreversibility on the appropriate decision criterion. Because explicit estimates of the likely rates of change of agricultural land use-values are rarely made and given the degree of uncertainty surrounding them, it is difficult to judge the comparability of the distribution used in this example with that which might be accepted as reflecting reality. The range adopted here, with a 60 per cent chance of the rate being within half of 1 per cent of zero, would not appear to be extreme. If the range is broadened to include growth rates of 3, 2, 1, 0, - 1, - 2, and - 3 per cent at probabilities of 0.1, 0 .1 , 0.2, 0.2, 0 .2 , 0.1 and 0.1 respectively with a time period of 20 years, the decision criteria become 2 . 5 with a discount rate of 5 per cent and 1.11 with a discount rate of 10 per cent. This still gives a 60 per cent chance of the actual growth rate being within 1 per cent of zero. The results are not dependent upon the mean of the distribution being zero or upon the distribution being symmetrical.

Conclusions The relevance of these numerical results depends upon the extent to which the key assumptions reflect reality. Those not dealt with in the sensitivity analysis relate to the complete irreversibility of the decision and to the zero cost of reversing the decision. The extent to which land use decisions are genuinely irreversible has been considered earlier. It is argued here that i t is not unrealistic to characterise a variety of such decisions in this way. The consequences of ‘irreversibility’ will be reduced where the costs of reversing the decision are less than the losses resulting from retaining the development use. The costs involved in reversing a decision, as opposed to those resulting from the delay in reaching the correct decision which have been accounted for, would involve the need to collect available data and to re-analyse the decision. While it is not possible to evaluate these factors, they tend to lead to an underestimate of the relative costs of choosing preservation. This could produce an overestimate of the optimal decision criteria. However, Ciriacy- Wantrup (1968) has suggested that planning agents generally have negative uncertainty preferences. If this is so, some degree of risk aversion could counter-balance the influence of the assumptions made.

UNCERTAINTY. IRREVERSIBILITY AND THE LOSS OF AGRICULTURAL LAND 20 1

It would seem that uncertainty as to the longer-run changes in the level of preservation values through time, which have been examined here, represent an additional argument for a conservative approach to decision-making to that associated with uncertainty due to the short-run, stochastic variation of values around the mean, such as incorporated in the analysis by Henry (1974) referred to earlier. Given this then, taken together, they indicate that development values should exceed preservation values by a greater extent in order to justify an irreversible change than would be indicated by either approach individually. Henry’s results, using a 10 per cent discount rate suggest that expected development values should exceed expected preservation values by around 13 per cent before development should be undertaken. This implies a decision criterion of 1.13. The results here, with a 20 year period to correct the decision, suggest a decision criterion of 1 .05. Taken together these would imply a maximum decision criterion of 1.19. Henry’s results with a 5 per cent discount rate indicate a slight increase to around 1.15, while the results here suggest a much greater increase to 1.25. Taking these two results together produces a maximum decision criterion of 1 .4.

Other assumptions have been made which simplify the model. These are that the value of development is known and constant and that the rate of change of preservation value is also constant. These assumptions are not preconditions of the consequences of irreversibility which have been demonstrated. Alternative interpretations of these variables could make the example less restricted. For instance, the rate of change in preservation value (g) could be taken as the difference between the change in development values and the change in preservation values and would be treated in the model in the same way. The value of development in year t (Q) could be made an expected value. This would increase the size of the expected maximum loss but, with indifference to risk, would not influence the optimal decision criterion.

Development values are commonly several times those generated by agriculture. The implications of this analysis are not therefore that, in general, the loss of land should be severely constrained on account of the irreversible nature of the change involved. However, the significance of the discount rate in this context has been demonstrated. Thus if, for instance, the arguments for lower discount rates (e.g., Price, 1973) were to be accepted, then the basis for preserving land in agricultural uses would become much stronger, applying to situations where the expected development benefits significantly exceed the expected agricultural benefits. The analysis does indicate that the optimal density of development, such as for housing, will be higher than would be indicated by studies ignoring the issue of irreversibility. Further, there will be occasions when, in a choice between alternative projects where one involves a significant permanent transfer of land from agriculture, such as in the choice between improving urban roads and constructing new rural ones, that the more conservative decision criterion could tip the balance in favour of preservation. Cost-benefit studies of irreversible land-use changes should be modified, or at the very least qualified, to take account of the significance of irreversibility. The final question of whether or not farm land should be protected turns not only on the issue of irreversibility, but also on an cva!uation of the other values of agricultural land, which were reviewed earlier in the paper.

These same principles apply equally to other irreversible changes, where future costs and benefits are uncertain. For instance, decisions relating to the protection of areas of natural or historical significance should take account not only of option and existence values but should also give further weight to protection to reflect the irreversible nature of the changes involved. The

IAN HODGE

evaluation of these issues is an area requiring further research.

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