UNIVERSITY OF NEWCASTLE N.S.W. AUSTRALIA

DEPARTMENT OF ECONOMICS

RESEARCH REPORT OR OCCASIONAL PAPER

Working Paper No. 98

Three Essays in Agricultural Economics

by

Clem Tisdell

May 1984

ISBN: 0 7259 486 0 [ISSN: 0812-1664]

UNIVERSITY OF NEWCASTLE, N.S.W., AUSTRALIA

DEPARMENT OF ECONOMICS

RESEARCH REPORT OR OCCASIONAL PAPER

Working Paper No. 98

Three Essays in Agricultural Economics1

by

Clem Tisdell2

May 1984

© All rights reserved

1 Essay I Non-Gradualism, Thresholds and Non-Convexities in Pest Control p. 1 Essay II Pest-control Levels, Adversities and Uncertainties p.11 Essay III Optimal Planting Rates for Crops Given Variability and Uncertainty about their Density of Establishment p.20

The Responsibility for the opinions expressed in the papers published in this series rests solely with the authors.

2 At the time of writing this paper, Clem Tisdell was Professor of Economics at The University of Newcastle,

NSW, Australia. He is now Professor Emeritus at the School of Economics, The University of Queensland, St. Lucia Campus, Brisbane QLD 4072, Australia. Email: c.tisdell@economics.uq.edu.au

1

ESSAY I

Non-Gradualism, Thresholds and Non-Convexities in Pest Control

ABSTRACT

Considers economic consequences for the optimal level of pest control at farm level of

discontinuities or jumps in the response of the pest population to control measures. In the

light of catastrophe theories, some account is also taken of hysteresis in control relationships.

It is shown that non-convexities in relevant production sets may arise from discontinuities in

the response of pest-populations to controls, as well as from other factors such as the

occurrence of thresholds. Consequently, a range of pest control measures or application rates

of controls can be ruled out as possible profit-maximizing ones, thereby simplifying the

problem of choosing an optimal pest-control strategy.

Keywords: Catastrophe theory, crop planting, pest control, risk and uncertainty.

JEL Classification: D81, Q01, Q10, Q21

2

Non-Gradualism, Thresholds and Non-Convexities in Pest Control

1. Introduction

While it was reasonable for the neoclassical economist Alfred Marshall (1930) to conclude

from the biological thought of his day, for example, from Darwin (1875) that nature does not

proceed by leaps and bounds but by gradual variations, current biological thought entertains

the possibility that jumps occur in biological phenomena, including the evolution of species

(Stanley, 1979; cf. Traub, 1983). Some of the relevant biological theory as applied to pest

control is based on catastrophe theory (Jeffers, 1978, Hollings, 1978; Zeeman, 1976; Thorn,

1975). Catastrophe theory recognizes discontinuities, hysteresis and unstable equilibria as

being important in biological relationships. This paper concentrates on the first of these

aspects.

The main purpose of this paper is to explore the implications for optimal economic levels of

pest control of non-gradual responses of pest populations to pest control measures, including

thresholds for responses. Phenomena with similar consequences for control policies, namely

non-convexities in response functions, are also considered and some attention is given to the

possibility of hysteresis in responses. The existence of thresholds and of non-convexities in

many response functions for pest populations is well established (Shoemaker, 1973; Heady,

1972) and evidence comes from the history of occurrence of Dutch elm disease (Jeffers,

1978) and from spruce bud worm infestations (Jones, 1977; Hollings, 1978) to support the

presence of discontinuities and hysteresis in some pest control responses. In practice,

economists need to be prepared for a range of properties in pest control responses including

the traditionally assumed continuous and smooth responses.

In order to concentrate on the main issues at hand, this paper deals with pest control at the

farm-level or level of the individual production unit and in. so doing ignores the presence of

externalities or spillovers (Tisdell, 1982, Ch. 9; Auld et al, 1978/79; Regev, et al, 1976; Feder

and Regev, 1975; Carlson, 1975). The paper is developed first by considering a catastrophe

model suggested by Jeffers (1978), then attention is given to the general possibility of non-

convexities, including thresholds, in pest response functions.

3

2. A Simple Catastrophe Model and Some of Its Economic Implications

Following Jeffers (1978), the nature of catastrophe theory can be illustrated by Figure 1

which shows a 'fold' case. Here y is the level of population of the pest and x is the

independent variable for controlling the pest. The variable x might be the application rate of a

chemical control or the level of release of population of a biologically controlling species, or

the amount of effort in mechanically removing the pest and so on.

The curve ABCDFG, which indicates the relationship between y and x, has two folds: one at

C and another at D. If x is increased from zero, the population of y follows ABC dropping at

extremity C (for a level of the control variable of x2) from y3 to y1 and for further increases of

x moves along FG. However, if x should be reduced after having been at or above x2, the

curve DF may be followed with a jump occurring at the extremity D (corresponding to x1)

and further reduction in x leads to movements along AB. This model has two 'catastrophe' or

discontinuity points.

The relationship may also be interpreted in the following way: if one is moving the pest

control variable from a position where x ≤ x1, the jump CF is relevant. If one is moving the

pest control variable from a position where x ≥ x2, the jump DB is relevant. This model

4

exhibits hysteresis as well as a discontinuity. "Hysteresis occurs when a system has an

apparently delayed response to a changing stimulus, and characteristically the response to the

stimulus follows one path when the stimulus increases and another path when it decreases".

(Jeffers, 1978, p. 149).

Figure 1 can also be adapted to illustrate a case in which a discontinuity in the pest-control

relationship is present but hysteresis is absent. Suppose that the relationship can be illustrated

by the curves ABC and FG. In this case a discontinuity (of CF) exists at x = x2 and at that

point, y drops from y3 to y1 as the magnitude of the pest-control variable is increased. The

pest-control process and its impact on the pest-population is perfectly reversible.

It is interesting to consider the economic implications for pest control of (a) a discontinuity in

the pest population/control function in the absence of hysteresis; (b) implications if a

discontinuity and hysteresis are present; and (c) if discontinuity is absent but hysteresis is

present. Let us examine case (a) first and then briefly consider the other two cases.

Given the relationship between y and x just described in which discontinuity without

hysteresis is present, a discontinuous functional relationship exists between the level of the

pest-control variable and the reduction achieved in the total population of the pest. As the

level of the control variable is increased the reduction in the population of the pest occurs

first at an increasing rate until x2 is reached, it then jumps in magnitude, and further

reductions occur at a decreasing rate. The relationship is like that shown in Figure 2, where r

represents the total reduction in the population of the pest. This relationship has some

similarity to the logistic one that is commonly believed to be relevant in relating percentage

mortality rates of an insect population to levels of insecticide application (Shoemaker, 1973;

Bullen, 1970) but differs from it because of the discontinuity.

5

The net profit from reducing the pest population as a result of increasing the pest control

variable x, depends upon the receipts from increased production less the cost of the control, x.

In theory, the most profitable level of control can be determined by considering iso-net profit

functions in the (x, r) space in Figure 2 and determining the population reduction/control

strategy that enables the firm to reach its highest attainable iso-net profit curve. However, it

may be more useful to consider a production function representation of the relationship.

The total additional production, z, resulting from an increase in the control variable x may

have the same form as the relationship shown in Figure 2 but need not have the same form. It

is a composite function. The additional production can be considered to be a function of the

reduction in the pest population, that is,

z = f (r) (1)

where the reduction in population is a function of ·the pest-control magnitude, that is,

r = h (x) (2)

which may have the form shown in Figure 2. Consequently, z as a function of x is,

z = f [h(x)] = g (x) (3)

6

Hence, by the chain rule,

dzdx

=f '. h' (4)

and is positive if both fʹ and hʹ are. By the product rule for differentiation,

𝑑2𝑧𝑑𝑥2

= 𝑓′.ℎ" + ℎ′.𝑓" (5)

If h" and f'" are both positive for x < x2 and negative for x > x2 the production function is of

the same form as that shown in Figure 2, assuming that the first derivatives are positive. A

production function of this type is shown in Figure 3 by curve ONPQR which has a

discontinuity at x2 where it jumps from P to Q.

If the price received for the product is a constant and the cost per-unit of the control is a

constant, the iso-profit lines relevant to Figure 3 are straight lines. In this case, profit is never

maximized by adopting a level of control between 0 and x2 that is, for values corresponding

to ONP boundary in which the set {x ≥ 0, z ≤ g (x)} is non-convex. Because of the

discontinuity or jump in the production response function, there is a range of values for the

pest control variable that are never optimal. In this case, the range 0 < x < x2 is always

inferior. The existence of this range is due to the non- convexity created in the production set

7

by the discontinuity or jump and the change in the sign of the second derivative of the

production function. In fact, a range of values of x that can never be optimal arise if either a

jump in the production function occurs or its second derivative changes its sign.

Given the form of the production response function shown in Figure 3, point Q is a corner-

point and the optimal value of x is likely to be either 0 or x2. Only if the price of the product

relative to the cost per unit of control is high enough, will a value of x greater than x2

maximize profit. Note that x2 corresponds to the level of pest control that maximizes

(additional) production per unit of the control. From the farmer’s strategic point of view, it is

important to locate this value of x, and for many profit-maximization problems it is the only

value of x that needs to be located if this type of response function applies. For advisers or

suppliers recommending a general application rate to farmers this rate would seem to be a

logical choice. However, the actual value of x for which the jump occurs may not always be

precisely known. In this case it is important to err on the safe side, that is, to err on the side of

a heavy application if application of the control measure is profitable.

It should be noted that even if the relevant iso-profit lines are non-linear that non-convexities

in the production set are likely to lead to ranges for the control variable that can never

maximize profit (Cf. Tisdell, 1982-83). These ranges depend upon the precise nature of the

objective function.

When hysteresis is present, the path by which the final optimal value of the control variable is

approached is important from a profit-maximizing point of view. The optimal course in some

cases may.be to overkill and then reduce application rates. However, hysteresis introduces

dynamic considerations and situations involving- it needs to be analysed by dynamic models.

Dynamic programming, for example might be applicable (Shoemaker, 1973; Feder and

Regev, 1975; Jones, 1977). If hysteresis and discontinuities are both present, not only will the

sequence of the control variable be important but there are likely ranges of the control

variable that will never maximize profit.

3. Non-Convexities in Continuous Cases and Thresholds

The population curve shown in Figure 2 can be regarded as an approximation to an S-shaped

curve which is believed to be applicable in many pest-control cases (Shoemaker, 1973;

Bullen, 1970). If an S-shaped continuous curve applies, the production possibility set also has

8

a non-convexity. The non-convexity occurs for those values of x less than the value of x for

which total reduction in the pest population per unit of control is at a maximum. If the

(additional) production function exhibits similar properties, the range of x-values that can

never maximize profit is similar to that identified in the last section.

The existence of thresholds (Headley, 1972; Shoemaker, 1973) can also be a source of non-

convexity in the pest-control production set. This is so even in cases where the production

function is continuous and does not have a segment rising at an increasing rate. Such a case is

illustrated in Figure 4 when OABC represents the production response to the pest control

variable, x. Up to a level of application there is no response but beyond x1 the response

occurs at a decreasing rate. Non-convexity clearly arises in the production set. If the iso-profit

line is linear, an application rate in the range 0 < x < x2, never maximises profit.

The existence of this threshold implies that even if fixed startup or overhead costs (a form of

economic lumpiness) are not involved in control, a range of values of the control variable, x,

are incapable of maximising profit. Thus the range of x-values requiring consideration from

an economic point of view is reduced and this is an aid to decision- making.

4. Conclusion

Non-convexities can arise in the relevant production sets associated with pest-control

9

measures because of discontinuities or jumps in the production response function, or because

this function has a positive second derivative over some of its range, or because thresholds of

control levels have to be achieved before a production response occurs. If the relevant profit

function or objective function is linear (and even in many cases where it is non-linear) this

non-convexity implies that there are a range of values for the pest control variable, x, that can

never be optimal. The decision-making problem is simplified by being able to delete these

values from consideration. The models discussed also indicate that there are rational

economic grounds for insensitiveness in the responsiveness of levels of pest-control

applications to price variations because of the importance of 'corner-points' in relevant

production functions.

5. Acknowledgement

I would like to thank Dr B. Auld for bringing my attention to the potential importance of

catastrophe theory for the economics of pest control and Mr W. Dunlop for his comments.

6. REFERENCES

Auld, B.A., Menz, K.A. and Monaghan, N.M. (1978/79), "Dynamics of Weed Spread:

Implications for Policies of Public Control", Protection Ecology, 1, 141-148.

Bullen, F.T. (1970), The Proceedings of the Ecological Society of Australasia, Vol. 5, pp. 63-

75.

Carlson, G.A. (1975), "Control of a Mobile Pest: The Imported Fire Ant", Southern Journal

of Agricultural Economics, 7(2), 35-41.

Darwin, C. (1875) On the Origin of Species by Natural Selection, Appleton-Century-Crofts:

New York.

Feder, G. and Regev, V. (1975) "Biological Interactions and Environmental Effects in the

Economics of Pest Control", Journal of Environmental Economics and Management, 2,

75-91.

Headley, J.C. (1972) "Defining the Economic Threshold" in Pest Control Strategies for the

Future, National Academy of Sciences: Washington, pp. 100-108.

10

Holling, C.S. (ed.) (1978) Adaptive Environmental Assessment and Management, John

Wiley: Chichester.

Jones, D.D. (1977), "The Application of Catastrophe Theory to Ecological Systems",

Simulation, 29 (1), 1-15.

Marshall, Alfred, (1930) Principles of Economics: An Introductory Volume, 8th ed.,

Macmillan: London.

Regev, U., Guttierrez, A.P. and Feder, G. (1976) "Pests as a Common Property Resource: A

Case Study of Alfalfa Weevil Control", American Journal of Agricultural Economics,

58, 186-197.

Shoemaker, C. (1973) "Optimization of Agricultural Pest Management", Mathematical

Biosciences, l6, 143-175; 17, 357-365; 18, 1-22.

Stanley, S.M. (1979) Macroevolution: Pattern and Process, W.H. Freeman & Co.: San

Francisco.

Thorn, R. (1975) Structural Stability and Morphogenesis, Benjamin: Reading, Mass.

Tisdell C.A. (1982) Wild Pigs: Environmental Pest or Economic Resource? Pergamon Press:

Sydney, New York.

Tisdell, C.A. (1982-83) "Pollution Control Policies Proposed by Economists", Journal of

Environmental Systems, 12 (4), 363-380.

Troub, R.M. (1983) "Economics and New Findings about the Biological Nature of

Humankind", Third World Congress of·Social Economics, Fresno (mimeo).

Zeeman, E.C. (1976) "Catastrophe Theory", Scientific American, 234, 65-83.

11

ESSAY II

Pest-Control Levels, Adversities and Uncertainties

ABSTRACT

Market or price adversity tends to lead to a reduction in the level of pest control but

technological (biological) adversity may result in increased levels of control. Although

expected utility maximisation combined with risk-aversion has been invoked to explain high

pest dosage rates, such dosage rates can occur under risk neutrality given the probable nature

of the marginal productivity function of pest control. Even under risk neutrality, increased

technological (biological) uncertainty about pest control measures is liable to raise pest

dosage rates. Consequences of this for public policy are noted.

12

Pest-Control Levels, Adversities and Uncertainties

1. Introduction

The purpose of this paper is to consider the impact on the level of private pest control, for

example, the amount of a pesticide or herbicide or effort applied in pest control, when the

economic value or effectiveness of the control diminishes or when uncertainty about its

economic value or effectiveness increases. Depending upon the source of increased adversity

or of increased uncertainty, the optimal level of private pest control can either increase or

decrease. Those impacts and what may appear to be an excessive level of pesticide

application or control effort by a landholder operating under conditions of uncertainty are not

dependent upon the presence of risk-aversion on the part of the landholder as some literature

may indicate (Feder, 1979; Norgaard, 1976). It is assumed that a private landholder wishes to

maximise his profit from pest control operations where certainty prevails, and under

uncertainty aims to maximise expected profit. The last assumption involves risk-neutrality

and is made to show that a number of effects that have been readily ascribed to risk aversion

in the literature could well have a different basis.

The article is developed by first considering reactions of pest control levels to various

adversities and then the impacts of uncertainties are introduced. The article concentrates on

private gains and strategies rather than social ones but some pertinent consequences for

public policy are drawn.

2. Adversities and Levels of Pest Control

Two main types of adversity may be encountered in pest control - market or price adversity

and or marginal productivity adversity. The former tends to lead to a reduction in the level of

pest control whereas the latter frequently results in an increase in the level of pest control.

Market adversity can result from a lowering of price of the product affected by a pest or a rise

in the marginal cost of applying the pest control. [Even though the latter is not entirely

market determined, it is convenient to include it under market factors]. An unfavourable

variation in the marginal productivity of the control can come about because of an

unfavourable environmental variation, e.g. a more resistant pest population, weather or

similar conditions that reduce the effectiveness of the control in lowering the pest population

13

and/or the response of the product to the pest reduction, or, in some circumstances, a greater

pest population. The consequences of such adversities can be seen by suitably modifying the

simple marginal productivity model for this case. In doing so, it is assumed that the mobility

of pests is not an important consideration in private pest control (Cf. Headley, 1972; Tisdell,

1982a, Ch. 9).

Assume that a landholder produces a single product which he sells at a price that he is unable

to influence and that his level of production of the product depends on the level of pest

control undertaken by him. The landholder's production function can be represented as

x = f (r) (1)

where x is the level of production and r is the level of pest control measured by a suitable

index. Consequently, the landholder's profit function is

π = px – C (r) (2)

= pf (r) – C (r) (3)

where p represents the price per unit of the product and C (r) is the cost of pest control. The

necessary condition for a maximum of this function is that

pfʹ(r) = Cʹ (r) (4)

that is, that the level of pest control be such that the value of marginal product from the

control equals the marginal cost of the control. In addition, the sufficient conditions for .a

maximum need to be satisfied. The profit-maximising condition is illustrated in Figure 1.

The line ABDE in this Figure represents the value of the marginal product of pest control

(conditions held constant) and the line JGK represents the marginal cost of pest control. The

profit-maximising level of pest control is r2. Note that in the case shown the value of

marginal product curve falls to zero and remains at this level for an application of r̅ or more.

(It is, of course, possible that it may become negative for very high applications).

Now consider the impact of a fall in the price of the product. This causes the value of

marginal product curve below r̅ to rotate anti- clockwise, the point D at r̅ being a fixed point.

[It is a fixed point because fʹ(r) = 0 for r = r̅ and hence no matter what the value of p, pfʹ(r)

14

= 0 for r = r̅]. In the case shown, for example, this segment rotates to FGD. Consequently, if

the marginal cost curve of control is not vertical, the optimal application rate of the control

falls. In the case illustrated it falls from r2 to r1. However, should the marginal cost of the

control be zero (or regarded by the landholder as virtually so) the optimal application rate

would be r and is not influenced by the price received for the product.

It is clear also that a shift upwards in the marginal cost of pest control, that is, in the curve

JGK, leads to a fall in the level of application necessary to maximise profit.

The impact of technological adversities in pest control is not so clearcut. One possibility is

that for environmental or other reasons pest control measures become less effective per

application unit in reducing the pest population. In that case, the value of marginal

productivity of control curve in Figure 2, ABD, may alter to a curve like LMNR, which

crosses ABD from below. Consequently, if the marginal cost of control curve falls below the

crossover point M, this adversity results in an increase in the profit-maximizing level of

application of the pest-control measure. On the other hand, should the relevant marginal cost

curve be above M, this adversity causes a fall in the optimal level of pest control. When the

marginal cost of pest control is low, this adversity can be expected to result in an increase in

the application of a pest control.

15

Other variations in marginal productivity of the control can also occur. A larger pest

population may cause the value of marginal product curve of control to shift to the right so

that a greater level of control becomes optimal, even though absolute profit is reduced due to

the greater population of the pest. Furthermore, a possibility of a shift to the left in the value

of the marginal productivity curve also exists, for example, if the response of the product,

e.g., a crop, to a reduction in the pest population becomes less. In this case, the optimal rate

of application of the control falls.

However, the above argument indicates that while market adversity can be expected to lead

to a reduction in the level of private pest control, various technological adversities in the

effectiveness of this control can result in a greater level of application of pest control

measures.

3. Uncertainties and Pest-Control Levels

Tendencies towards what appear to be excessive levels of application of pesticides

(applications considerably exceeding those optimal for the expected or most probable state of

nature) have often been ascribed to risk aversion on the part of landholders (Feder, 1979,

Norgaard. 1976). However, even when expected profit maximisation is the objective of a

landholder, the nature of the marginal productivity curve of pest control may be such that

excessive application rates (in the above sense) are likely. This is especially so for

technological uncertainty, given that value of marginal product curves fall to zero after some

16

rate of pesticide application and remains there, but is unlikely for price uncertainty. This can

be most readily seen if the marginal cost of pest control is assumed to be zero.

In this case, increased uncertainty about the price of the product has no effect on the level of

application of the control. If the value of the marginal productivity relationships are of the

form indicated in Fig. 1, the optimal control level remains at r̅, no matter what is the

probability distribution of the price of the product. This is because pfʹ(r) = 0 for r = r̅ no

matter what is the value of p because fʹ(r̅) = 0.

However, increased ‘technological’ uncertainty is likely to increase the level of pest

application to maximise expected profit) because there is a chance that the value of marginal

product becomes positive for higher pest applications than those levels optimal with less

uncertainty. In fact, given the zero marginal costs of pest control, the level of pest control

necessary to maximise expected profit is the largest rate (value of r) necessary to equate

every possible value of marginal productivity curve to zero. This can be illustrated by Figure

3.

Let AJBD represent the value of the marginal product of pest control under certainty and

compare this with a circumstance in which EFD and GHD are each probable with a

probability of 0.5. Under certainty, the optimal application rate is r2. Under uncertainty, the

value of the expected marginal product curve is AJHD, given that EA = AG and EF is

parallel to GH. Consequently under uncertainty the optimal application rate of the .pest

control increases to r3. Even if the marginal cost of pest control is positive; increased

technological uncertainty will induce an increase in the rate of application of the control

provided that the marginal cost curve lies below J. The lower is the marginal cost of pest

control, the more likely is technological uncertainty to lead to an increase in the level of pest

control.

17

Even if the impact of technological uncertainty on the value of marginal product curves of

pest control is not symmetrical (even if these curves cross over), this uncertainty is likely to

increase the r-value required for the value of marginal product of pest control to be definitely

zero and raise the optimal level of control given that marginal productivity of control falls to

and remains at zero for applications at or exceeding some threshold. In fact, given zero

marginal rates of pest control and the assumed nature of a marginal productivity function,

expected profit is maximised by adopting the largest application rate that is optimal for any

state of nature in the set of possibilities. This bias towards 'excessive' application is not a

consequence of risk- aversion. Such a 'bias' can be expected to continue even where low

levels of marginal cost of pest control are encountered.

4. Conclusions and Qualifications

The simple theoretical considerations outlined above indicate that market adversities are

likely to lead to a reduction in the level of pest control by a landholder, e.g., a farmer, but

that, by contrast, technological (biological) adversity or increased uncertainty is likely to

result in a greater level of pest control. It has also been emphasized that 'excessive'

application of pesticides can be explained by the possible nature of the marginal productivity

curve of pest control. This is assumed to be zero for application rates at or above a particular

threshold.

18

Should the marginal product of the pest application move quickly from being zero to being

negative as the application is increased, the above theory would need qualification as far as

the discussion of the impact of uncertainty is concerned. However, if a zero marginal

productivity value is maintained for some range of application or if the possible negative

marginal productivity of increased application of pest control is very small, the substance of

the above argument still continues to apply. However, the assumption that the total revenue

productivity of a pest-control measure (given constant product prices) reaches a constant and

remains stationary above a threshold level of application (which of course implies zero

revenue marginal productivity) or closely converges to such a constant, appears to be

accepted in the relevant ecological literature (Bullen, 1970, esp. p. 67) as are closely

associated S-shaped percentage pest-population kill functions for pesticide dosage rates

which reach or approach an upper stationary value as dosage rates increase (Shoemaker,

1973, esp. p. 5; Bullen, 1970. esp. p. 65).

The above theory also suggests that when a pesticide is being applied by landholders in

quantities that are considered to be excessive from a social point of view, it may be possible

for the government to reduce application rates on average by providing landholders with

additional information that reduces the extent of their technological (biological) uncertainty

about the private productivity of pest control measures. Excessive private application of pest

controls from a social viewpoint may be bound up with a variety of factors including

externalities and these aspects have been widely canvassed in the literature (Tisdell, Auld and

Menz, 1984; Tisdell, 1982b; Feder and Regev, 1975; Taylor and Headley, 1975; Hueth and

Regev, 1974; Langham and Edwards, 1969). The paper therefore has some public policy

relevance as well as providing an alternative or supplementary explanation for 'excessive'

application of pesticides by private individuals. In particular the consequences are not

dependent on the two general hypotheses about individual behaviour invoked in an earlier

explanatory model presented by Feder (1979): maximisation of expected utility and risk

aversion.

5. References

Bullen, F.T. (1970) "Benefit/Cost Analysis of Various Degrees of Crop Protection",

Proceedings of the Ecological Society of Australia, 5, 63-75.

Feder, G., (1979) "Pesticides, Information and Pest Management under Uncertainty",

19

American Journal of Agricultural Economics, 6, 97-103.

Feder, G. & U. Regev (1975) "Biological Interactions and Environmental Effects in the

Economics of Pest Control", Journal of Environmental Economics and Management, 2,

75-91.

Headley, J.C. (1972) "Defining the Economic Threshold", Pest Control Strategies for the

Future, National Academy of Sciences: Washington, D.C., pp. 100-108.

Rueth, D. and U. Regev (1974) "Optimal Agricultural Pest Management with Increasing Pest

Resistance", American Journal of Agricultural Economics, 56, 543-541.

Langham, M.R. and W.F. Edwards (1969) "Externalities in Pesticide Use", American Journal

of Agricultural Economics, 51, 1195-1201.

Norgaard, R.V. (1976) "The Economics of Improving Pesticide Use'', Ann. Rev. Entomol., 21,

45-60.

Shoesmith, C. (1973) "Optimization of Agricultural Pest Management III: Results and

Extensions of a Model", Mathematical Biosciences, 18, 1-22.

Taylor, C.R. and J.C. Headley (1975) "Insecticide Resistance and the Evaluation of Control

Strategies for an Insect Population", Canad. Entomol., 107, 232-242.

Tisdell, C.A. (1982 a) Wild Pigs: Environmental Pest or Economic Resource? Pergamon:

Sydney, New York.

Tisdell, C.A. (1982 b) "Exploitation of Techniques that Decline in Effectiveness with Use",

Public Finance, 37, 428-437.

Tisdell, C .A., B. Auld and K. Henz (1984) "On Assessing the Value of Biological Weed

Control", Protection Ecology, 6, 169-179

20

ESSAY III

Optimal Planting Rates for Crops Given Variability and Uncertainty

About Their Density of Establishment

ABSTRACT

Examines, on the basis of general principles, the optimality of 'overplanting' of propagative

material (seeds, seedlings, tubers, etc.) for crops given uncertainty and variability in their

establishment rates and taking account of the functional relationship between the yields of

many crops and their established densities. It is shown to be optimal in most cases to plant

more material than would maximize yield should the most favourable anticipated state of

nature prevail.

21

Optimal Planting Rates for Crops Given Variability and Uncertainty

About Their Density of Establishment

1. Introduction

The influence of the population-density of crops on their yields is well documented (Willey

and Heath, 1969). For many crops the quantity of yield as a function of the established

density of the crop rises to a single relatively sharp peak with increased density and declines,

but in some cases the relationship between population-density and yield has a flat-topped

peak or increases asymptotically (Roberts, 1972). These relationships are not only important

for the level of yield obtained by the grower of a crop but also for the profit (and utility) that

he obtains from his operations.

In practice, the density at which a crop is established is not a perfectly controlled variable but

is subject to environmental variability and uncertainty. While the establishment .rate of a crop

usually depends on the amount of planted regenerative material, e.g., seeds, seedlings, tubers,

etc., environmental and related factors operating on this botanical material both prior and

after planting also affect the yield (Austin, 1972; Christensen, 1972; Pollock, 1972; Moore,

1972). These environmental and related factors are variable and the individual grower is not

in a position to have perfect information about them or to predict them perfectly. This is so

even when seed lots have been certified to satisfy a germination level greater than a specified

level because germination tests are done in the laboratory under ideal conditions and

significant discrepancies have been found between these results and the emergence of seeds

in the field (Austin, 1972; Perry, 1967; Austin, 1963).

Roberts (1972, p. 307) has observed in relation to seeds that:

"The deterioration leading to the loss of viability in seeds can affect the yield of a

crop in two ways: first the decreased germination can lead to a sub-optimal

population of plants per unit area; secondly, the deterioration - of which seed

viability may be an index - may result in a poorer performance by the surviving

plants".

He points out that one can compensate for both effects by increasing the seeding rate. To do

so in an optimal manner would be relatively straightforward if one had perfect information.

22

However, in most practical situations facing growers, perfect knowledge is lacking.

The relevant decision-making problem for growers is to determine the optimal amount of

regenerative material to plant given that the quality of the propagative material is to some

extent uncertain and that its establishment in the field will be influenced by such random

variables as weather conditions and pest attacks.

The purpose of this paper is to consider this decision-making problem by means of a simple

model in which it is assumed that there is no thinning or infilling of the crop after its

emergence or establishment. The problem is to determine the optimal planting rate given

variability and uncertainty about the establishment rate of the crop.

2. A Simple Model

Suppose that the grower aims to adjust the planting rate of his crop so as to maximize its

expected yield per unit of area, that is, to maximise its yield on average per hectare. (A

similar mathematical argument applies to that given below if the grower aims to maximise his

expected profit or utility from the crop rather than its yield). Suppose that the yield, y, of the

crop depends on the density, x, established for the crop (Willey and Heath, 1969, Tisdell and

De Silva, 1984) and that the established density of the crop is a function of the rate of

planting of the relevant type of regenerative material. Mathematically in this non-random

relationship

y = f (x) (1)

x = g(r) (2)

where, for example, expression (1) might have the form illustrated by Figure 1. In order to

maximize yield it is necessary to select r so that

dydr

=f '.g'=0 (3)

The smaller is gʹ, the higher is the planting rate needed to ensure a particular establishment

rate. If x = br then a planting rate of x/b is needed to ensure an establishment rate of x. If the

establishment rate maximising yield is x�, then a planting rate of x�/b is needed to attain the

optimal density of the crop. This, however, assumes that there is a clearcut functional

23

relationship between the planting rate and the establishment rate of the crop. The most

common situation is one in which the relationship between the planting rate and the

establishment rate is to some extent random.

To model this situation let us suppose that two alternative states of nature are possible and

that if the more favourable state prevails, the establishment rate of the crop is

xu = v (r) (4)

but if the worst state of nature prevails it is

xm = w(r). (5)

In many cases, xm can be expressed as a function of xu. Assuming that this is possible, we can

express the relationship by

xm = h(xu). (6)

Given that the probability of the most favourable state of nature is p, that of the least

favourable is (1 - p), the expected yield from the crop can be expressed as

E[y] = pf(xu) + (1 - p) f (xm) (7)

24

= pf(xu) + (1 - p) f [h(xu)]. (8)

A necessary condition for expected yield to be maximised is that.

dE[y]dxu

=pf '(xu)+�1-p�f '(xu)h'(xu)=0 (9)

There may be cases, for example, in which (6) is of the form·

xm = kxu (10)

and therefore hʹ(xu) in (9) is equal to k. In any case, xm might normally be expected to

increase with xu, that is, the relationship hʹ(xu) > 0 might be expected to hold.

By considering expression (9), we can determine the way in which the optimal value of xu, x�,

and the associated optimal planting rate, r�, differs as a result of uncertainty about the

establishment rate from the optimal rates, x� and r̅, where there is no uncertainty about

establishment rates. As will be shown in the next section, it is optimal to undertake extra

planting to allow for the uncertainty of establishment of the crop (the last condition being

taken as a reference point).

3. Some Implications of the Model

Expression (9) is satisfied if

(1 - p) fʹ (xu) hʹ (xu) = -pfʹ (xu). (11)

The left-hand side of this expression represents the rate of change in expected yield resulting

from an increase in xu (and, therefore, in xm if hʹ > 0) if the worst state of nature prevails

whereas the right-hand side can be interpreted as the rate of change in the expected penalty

(or cost) in terms of yield forgone by increasing xu if the best state of nature prevails.

Assuming that the yield·function y = f(x) increases at a decreasing rate, is unimodal and has a

unique maximum at x� (for example as illustrated by Figure 1), a curve like .LMN in Figure 2

corresponds to the expression on the left-hand side of (11) and a curve like PRS corresponds

to that on the right-hand side of (11).

25

Now, at x� curve LMN must always exceed PRS assuming that xm < xu because for x < x�,

fʹ(x) is positive and increasing xu raises xm and makes a positive marginal contribution to

yield. Hence, LMN and PRS must intersect to the right of x� and consequently the value of x,

x�, satisfying (11) exceeds x�. It also follows that r� > r̅, that is, to cope with the uncertainty one

should 'overshoot' in the planting rate compared to the optimal planting rate if the best state

of nature prevails.

Some other observations can also be made by reference to Figure 2:

(a) The greater is the probability of the worst state of nature, the greater 1 - p, and

therefore the lower the probability of the best state of nature, the greater is x� and

r�, the higher is the optimal overshoot in planting rates. This follows since R and

N are fixed points in the sense that they are independent of p. For instance, it

follows that if fʹ(xu) = 0 for x� so does pfʹ(xu).

(b) The more slowly PRS rises after x� relative "to the decline in LMN the greater is

the optimal overshoot in the planting rate. For example, other things equal, the

26

slower is the rate of decline of f(x) after reaching its maximum the greater is the

optimal overshoot.

(c) Other things equal, the smaller xm tends to be in relation to xu, the greater is the

optimal overshoot, given that f(x) increases at a decreasing rate. The smaller is xm

the greater is the marginal contribution to yield to be expected by increasing it.

4. Qualifications and Conclusions

This paper has identified circumstances in which yields on average can be increased by

raising planting rates to counteract the risk of crop establishment failures. In particular, it

isolates factors that influence the optimal degree of 'excess' planting when uncertainty and

variability in establishment rates are anticipated.

It is a rather simplified illustration of the factors at work. While it only considers two

alternative possible states of nature, similar conclusions follow if more than one state of

nature is possible, for example, the desirability of planting at a greater rate than would be

optimal should the most favourable state of nature prevail. Furthermore, the rate of crop

failure is assumed to be uniform in the field. In practice, it may be patchy. Yet even with a

degree of patchiness in the crop the general qualitative conclusions will continue to hold.

The possibility of systematic crop infilling or thinning after the establishment of the crop (Cf,

Tisdell and De Silva, 1984) has been ignored. To the extent that this strategy is possible and

costs little, it will reduce the desirability of overshooting in planting rates to compensate for

environmental risks.

It is clear that given the state of knowledge about yield/density planting functions for various

crops that it would not be difficult in particular instances to determine the planting rate

required to maximise expected yield from a crop given the main environmental risks

anticipated for the crop. Planting strategies that do not take these environmental risks into

account· may result in considerable losses in crop production on average.

5. Acknowledgement

I would like to thank Mr N.T.M.H. de Silva for bringing my attention to some relevant

material dealing with this problem.

27

6. References

Austin, R.B. (1972) "Effects of environment before harvesting on viability", pp. 114-149 in

E.H. Roberts (ed.) Viability of Seeds, Chapman and Hall: London.

Austin, R.B. (1963) "Yield of onions as affected by place and method of seed production",

Journal of Horticultural Science, 38, 277-85.

Christensen, C.M. (1972) ''Microflora and seed deterioration", pp. 59-93 in E.H. Roberts (ed.)

Viability of Seeds, Chapman and. Hall: London.

Moore, R.B. (1972), "Effects of mechanical injuries on viability" pp. 94-113 in E.H. Roberts

(ed.) Viability of Seeds, Chapman and Hall: London.

Perry, D.A. (1967) "Seed vigour and field establishment of peas", Proceeds of the

International Seed Testing Association, 32, 3-12.

Pollock, B.M. (.1972) "Effects of environment after sowing on viability", pp. 150-171 in E.H.

Roberts (ed.) Viability of Seeds, Chapman and Hall: London.

Roberts, E.H. (1972) "Loss of viability and crop yields", pp. 307-320 in E.H. Roberts (ed.)

Viability of Seeds, Chapman and Hall: London.

Tisdell, C.A. and De Silva, N.T.M.H. (1983), "The economic spacing of trees and other

crops", European Review of Agricultural Economics, 10, 281-293.

Willey, R.W. and Heath, S.B. (1969) "The quantitative relationships in plant population and

crop yield", Advances in Agronomy, 21, 282-321.

28

PREVIOUS PUBLICATIONS IN THIS SERIES (TO 1984)

1. JOHNS, B.L., "Import Substitution and Expert Potential - The Case of Manufacturing Industry in West Malaysia", October 1973, ISBN 0 7259 0063 6. - Also published in Australian Economic Papers, 12(21), December 1973, pp. 175-195.

2. JACOBI, S.N., "Economic Policy Alternatives for Relieving Urban Traffic Congestion", October 1973, ISBN 0065 2. - Also published in Webb, G.R. & J.C. McMaster, (eds.) Australian Transport Economics,

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9. IP, P.C., "Inflation, Unemployment and Economic Growth", June 1974, ISBN 0 7259 0074 1. 10. DUNLOP, W.C., "Banana Marketing", July 1974.

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29

22. TISDELL, C.A., "Promotion and Advertising by Monopolies and Cartels - A Neglected Welfare Aspect", November 1975, ISBN 0 7259 0212 4. - Also published as "Is Advertising Expenditure Socially Excessive?", in Bulletin of

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30

41. TISDELL, C.A., "Observations on the Wild Pig Problem in N.S.W. - A Survey and Interpretation of Economic Aspects based on Reports from Pasture Protection Boards", March 1978, ISBN 0 7259 0 304 X. - Incorporated in Wild Pigs: Environmental Pest or Economic Resource? (Pergamon Press,

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48. TISDELL, C.A., "On the Economics of Saving Wildlife from Extinction", February 1979, ISBN 0 7259 0329 5.

49. SHARPE, I.G. & VOLKER, P.A., "The Australian Reserve Base/Money Relationship", May 1979, ISBN 0 7259 0345 7. - Also published as "The Australian Monetary Base/Money Supply Relationship 1964-1977",

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52. MATHEWS, R.L., "The Distribution of Tax Sharing Entitlements Among the States", October 1979, ISBN 0 7259 0362 7, (The Fourth Newcastle Lecture in Political Economy).

53. OAKLEY, A.C., "The Value-Price-Distribution Articulation Problem in Karl Marx's Critique of David Ricardo's Principles", May 1980, ISBN 0 7259 0378 3. - A revised version appears as Chapter 4 in Marx's Critique of Political Economy: Intellectual

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57. TISDELL, C.A., "Law, Economics and Risk-Taking", October 1980, ISBN 0 7259 0393 7. - Also published in Kyklos, Vol 36 No l, 1983, pp. 3-20.

58. FISHER, J R., "Tenurial Deficiencies in the English Land System: The Mid-Nineteenth Century Debate", November 1980, ISBN 0 7259 0397 X. - An amended and abbreviated version appears in Agricultural History Review 31, Part 1,

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59. COATS, A.W., "Reflections on the Professionalization of Economics", November 1980, ISBN 0 7259 0399 6.

60. UHR, C G., "Notes on the Influence of Wicksell's Theories on American and British Economic Thought", July 1981, ISBN 0 7259 0411 9.

61. STAHL, C.W., "International Labour Migration and International Development", August 1981, ISBN 0 7259 0411 9.

62. KEATING, G.R. & SHARPE, I.G., "Australian Interest Rates: A Cross Correlation of Analysis" September 1981, ISBN 0 7259 0413 5. - Also published in Jüttner, D.J. (ed.) Interest Rates, (Longman Cheshire, Melbourne, 1981)

pp. 181-203. 63. TISDELL, C.A., “The Patent System: An Economic Review Concentration on the Life of

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64. GORDON, B.L.J., "Studies -in the Economics of W.S. Jevons: A Centenary Checklist", December 1981, ISBN 0 7259 0419 4.

65. TISDELL, C.A., "Resource Allocation and Control Over Man's Environment: Three Economic Essays", March 1982, ISBN 0 7259 0424 0. - Essay I also published in Environmental Systems, 12(2), 1982-83, pp. 153-161; Essay II in

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66. TISDELL, C.A., "Oligopoly and the Impact of Variable Demand Conditions on Profit and the Flexibility of Techniques", April 1982, ISBN 0 7259 0425 7.

67. TISDELL, C.A., "Production and the Natural Environment: Two Economic Essays", April 1982, ISBN 0 7259 0427 5. - Essay II also published in Journal of Agricultural Economics, 34(2), 1983, pp. 175-185.

68. PULLEN, J.M., "The Balanced Budget Multiplier Theorem: Some Comments on its History, and a Critique", June 1982, ISBN 0 7259 0431 3.

69. SHARPE, LG., "New Information and Australian Equity Returns: A Multivariate Analysis", June 1982, ISBN 0 7259 0432 1. - Also published in Australian Journal of Management, 8(1), June 1983.

70. DOELEMAN, J .A., "Concerning the Conflicting Nature of the Contribution of the Discipline of Economics to the Teaching of Environmental Studies", July 1982, ISBN 0 7259 0436 4. - Also published in Journal of Environmental Education, forthcoming, 1983.

71. UHR, C.G., "The Economic Writings of Sir William Petty, 1623-1687, Revisited", July 1982, ISBN 0 7259 0435 6.

72. SHARPE, I.G., "On the Predictability of the Spot U.S.$/A$ Exchange Rate:1978-1981", July 1982, ISBN 0 7259 0437 2.

73. TISDELL, C.A., "The World Conservation Strategy: Its Economic Basis and Australian Proposals", August 1982, ISBN 0 7259 D440 2. - Also published as "An Economist's Critique of the World Conservation Strategy, with

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74. JACOBI, S.N., "The Economics of Crime: A Survey of Issues", August 1982. ISBN 0 7259 0441 0.

75. SHARPE, LG. & HOGAN, W.P. “Regulation, Investor/Depositor Protection and the Campbell Report” June 1982, ISBN 0 7259 0444 5. - Also published as "On Prudential Controls", in Economic Papers Special Edition on The

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76. TISDELL, C.A., "Three Microeconomic Essays", September 1982, ISBN 0 7259 0445 3. - Essay I also published in The Manchester School of Economic and Social Studies, 51(2),

1983, pp. 152-158; & Essay II in Oxford Agrarian Studies, forthcoming

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77. TISDELL, C.A. & FAIRBAIRN, I .J., "Subsistence Economies and Unsustainable Development and Trade: Some Simple Theory", September 1982, ISBN 0 7259 0446 1. - Also published in The Journal of Development Studies 20(2), January, 1984.

78. SHARPE, I.G., "The Treasury Note Tender and Volatility of Australian Short-Term Interest Rates", October 1982, ISBN 0 7259 0447 X.

79. TISDELL, C.A. & DE SILVA, N.T.M.H., "Economic Spacing of Trees and Other Crops", November 1982, ISBN 0 7259 0448 8. - Also published in European Review of Agricultural Economics, 1983, 10(3), pp. 281-293.

80. SHARPE, I.G., "Covered Interest Rate Parity: The Australian Case", March 1983, ISBN 0 7259 0452 6. - Also published in Applied Economics, forthcoming 1984.

81. FISHER, J.R. & SMITH, A., "Tariffs and the Victorian Wire Industry in the Federation Era", April 1983, ISBN 0 7259 0453 4.

82. TISDELL, C.A. & FAIRBAIRN, I.J., "Development Problems and Planning in a Resource-Poor Pacific Country: The Case of Tuvalu", April 1983 ISBN 0 7259 0454 2. - Also published in Public Administration and Development, forthcoming.

83. SHARPE, I.G. & HOGAN, W.P., “On the Relationship Between the New York Closing Spot US $/$A Exchange Rate and the Reserve Bank of Australia’s Official Rate”. June 1983. ISBN: 0 7259 0456 9 - Also published in Economic Letters, forthcoming 1983.

84. FORSTER, B.A., “Acid Rain in North America: An International Externality”, July 1983. ISBN: 0 7259 0458 5.

85. TISDELL, C.A. AND FAIRBAIRN, I.J., “Labour Supply Constraints on Industrialization and Production Deficiencies in Traditional Sharing Societies”, August 1983, ISBN: 0 7259 0461 5

86. GORDON, B.L.J., JARVIE, W. & GORDON, M. “Sub-Regional Labour Markets in Newcastle and the Hunter: Part One, the 1971 Census”. September 1983, ISBN: 0 7259 0466 6.

87. DICK, H.W., “PLUS CA CHANGE … The Evolution of Australian Liner Shipping Policy”, October 1983, ISBN: 0 7259 0467 4.

88. GRUEN, F.H., “The Prices and Incomes Accord, Employment and Unemployment”, September, 1983, ISBN: 0 7259 0469 0 (The Seventh Newcastle Lecture in Political Economy).

89. KIBRIA, M.G. & TISDELL, C.A., “Productivity Progress and Learning by Doing in Bangladesh Jute Weaving Industry'', October 1983. ISBN 0 7259 0470 4.

90. McSHANE, R.W. & SHARPE, I.G., "A Time Series/Cross Section Analysis of the Determinants of Australian Trading Bank Loan/Deposit Interest Margins:1962- 1981", October 1983, ISBN 0 7259 0471 2.

91. TISDELL, C.A., "Cost-Benefit Analysis, The Environment and Informational Constraints in LDCs", November 1983, ISBN 0 7259 0472 0.

92. KIBRIA, M.G. & TISDELL, C.A., "Inflexibility of Industrial Employment in a Third World Country: The Case of Jute Weaving in Bangladesh”, November 1983, ISBN 0 7259 0473 9.

93. GORDON, B. & JOSEPH, E., "Studies in the Thought of Joseph A. Schumpeter, Economist: A Centenary Checklist", November 1983, ISBN 0 7259 0474 7.

94. PULLEN, J.M., "Malthus, Jesus, and Darwin", January 1984, ISBN 0 7259 0476 3. 95. TWOHILL, B.A., AISLABIE, C.J. & SHEEHAN, W.J., “The Concentration Phenomenon and

Stability Problems in a Micro-Economy: The Norfolk Island Public Sector Experience, 1976-77 to 1982-83” March 1984, ISBN 0 7259 0483 6.

96. FISHER, J.R., “Australia and the First Economic Revolution”, April, 1984, ISBN 0 7259 0484 4.

97. TISDELL, C.A., "Two Essays in Managerial Economics”, May, 1984, ISBN 0 7259 0485 2. 98. TISDELL, C.A., “Three Essays in Agricultural Economics", May, 1984, ISBN 0 7259 0486 0. 99. KEATING, G., "State Lottery Subscriptions - An Analysis Using Spline Regression”, May

1984, ISBN 0 7259 0488 7. 100. STANTON, P.J., “Protection and Structural Adjustment in the Australian Tyre Industry, 1960

to 1980”, June 1984, ISBN: 0 7259 0489 5. 101. TISDELL, C.A., “Externalities and Coasian Considerations in Project Evaluation: Aspects of

Social CBA in LDCs”, June 1984.

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102. DOELEMAN, J.A., “Historical Perspective and Environmental Cost-Benefit Analysis”, July 1984, ISBN: 0 7259 0492 5.

103. POWELL, A.A., “Real Wages and Employment”, July 1984, ISBN: 0 7259 0494 1. (The Eighth Newcastle Lecture in Political Economy)

104. TISDELL, C.A., “Costs and Benefits of Tree Conservation, Maintenance, Regeneration and Planting: Evaluation of Case Studies”, August 1984, ISBN: 0 7259 0495X