Agricultural & Applied Economics Association
On the Timing and Application of Pesticides Author(s): Darwin C. Hall and Richard B. Norgaard Source: American Journal of Agricultural Economics, Vol. 55, No. 2 (May, 1973), pp. 198-201 Published by: Blackwell Publishing on behalf of the Agricultural & Applied Economics Association Stable URL: http://www.jstor.org/stable/1238437 Accessed: 23/04/2009 15:23
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http://www.jstor.org On the Timingand Applicationof Pesticides*
DARWIN C. HALL AND RICHARD B. NORGAARD
The modeldeveloped by J. C. Headleyto illustratethe entomologists'concept of the economic thresholdis presentedand criticized.A two-variablemodel directed at the problemsof optimal timingand pesticideapplication as well as optimalpest populationlevel is presented.
FOR economic, ecological, and social rea- in dollars to pest density and time. The eco- sons, preventative spraying with heavy nomic threshold is that population level where doses of nonselective, persistent insecti- the marginal benefit from damage prevented by cides is becoming obsolete. Pest resistance, the control program is equal to the marginal rapid pest resurgence due to a lack of natural cost of realizing that population through a con- control by predators recently killed from appli- trol program. The model has four basic ele- cation of pesticides, socially unacceptable en- ments: a pest population growth function vironmental costs, and other phenomena result- (equation 1.1), a pest damage function (equa- ing from preventative spraying have gradually tion 1.2), a product yield function (equation led to more remedial spraying with selective 1.3), and a pest control cost function (equation pesticides. Entomologists are now promoting 1.4). Each of these elements is presented below. integrated control, i.e., using the best combina- = tion of all known inputs and techniques includ- (1.1) Pt Pt_-(l + r)n ing biological controls, cultural practices, and (1.2) Dt = bPt2 - A chemical approaches. The philosophy of inte- Y = N - cDt grated control is well established; but, due to (1.3) the disaggregated approach of past entomologi- h cal research and the limited attention (1.4) K= paid by Pt-n economists to the problem, methods are still ad hoc. The dichotomy between the state of the where and that of the practice is illustrated philosophy level at time by considerable concern over the "economic Pt= pest population period t, the harvest threshold," a term used by entomologists to de- time; level n note the pest population level at which controls Pt_n=pest population periods prior to should be initiated [3, p. 240]. Entomologists t; r= net rate of the advocate using all inputs in their best combina- growth pest population time tion and simultaneously admit considerable un- per period; D= cumulative at time 1, certainty as to when and how even a single damage period as a function of where the control input, such as an insecticide, should given Pt, be used. pest population has grown from Pt-, to Pt with no external interruption; The Headley Model A = a constant related to the pest damage The definition of the economic threshold has tolerance level based on the recuperat- been investigated by J. C. Headley ing potential of the crop; recently = within the framework of a simple pest popula- b a parameter relating units of pest pop- of tion growth model and a single application of a ulation to units crop damage; at harvest time pesticide [2]. The model relates crop damage Y= realized product yield in * (t) dollars; Giannini Foundation Paper No. 359. The authors are at harvest time if no Darrell N=potential yield indebted to Gerald A. Carlson, J. C. Headley, in Hueth, and an anonymous reviewer for their constructive pest damage, expressed dollars; comments on earlier drafts of this paper. The work reported c=a parameter relating damage units to here was funded in part by an IBP sponsored project en- dollars; titled, "The Principles, Strategies, and Tactics of Pest K total cost in dollars of reducing the and Control in Eco- Population Regulation Major Crop to Pt-n at time t-n; systems." pest population and DARWIN C. HALLis a research assistant of agricultural h= a parameter relating the inverse of economicsand RICHARDB. NORGAARDis assistant professor units to dollar units of con- of agricultural economics at the University of California, population Berkeley. trol costs. 198 May 1973 TIMING AND APPLICATIONOF PESTICIDES / 199
Combining the first three equations yields: Pest Population Level (1.5) Y = N - c{b[Pt.n(l + r)n]2 - A}.
Pt-n- The marginal change in yield due to an incre- mental increase in the pest population level at time period t-n is: I dY I (1.6) = - 2cb(1 + r) 2Pt-n. dPt-n I I The marginal change in the cost of pest con- I Ime trol due to an incremental change in 0 tht-n t pest popu- (planting) (application of (harvest) lation at time period t-n is: pesticide) dK h Figure 1. Pest populationlevels over time (1.7) dPt-n Pt_-,2 the effectiveness of most pesticides is fairly well Equating marginal revenue to marginal cost documented. The important issues are con- determines the optimal level to which the popu- cerned with when they should be used. lation should be reduced during time period Second, the cost of control must be a function t-n: of the level of population before controls are initiated, Pt-n-a, and the difference between / h \1/3 Pt-n-t- and the level reached after control, Pt-n. (1.8) Pt-, 2cb(1 + r)2n Since Pt-n-^ is a function of Po, the pest popula- tion at the time of planting, Headley's cost This is Headley's economic threshold. It indi- function is valid only for a unique, unspecified cates the level to which the pest population value of Po. It should be noted, furthermore, should be reduced during time period t-n such that, if the time of pesticide application t-n is a that the damage due to the growing pest popu- variable, the cost function must be considerably lation between t-n and harvest time t is mini- respecified. mized to the cost of control. subject pest A Two-Variable Model Headley's model nicely illustrates several of the factors which must be considered in any The following model corrects the above de- definition of the economic threshold. Neverthe- ficiencies yet retains as much as possible of the less, it has several limiting assumptions. First, simple general form of Headley's model. It con- Headley implicitly assumes that the period of sists of five elements: a pest population growth pesticide application t-n is "entomologically function (equation 2.1), a pest population kill determined."' Thus pest damage prior to t-n function (equation 2.2), a pest population dam- cannot be controlled and is not considered in age function (equation 2.3), a product yield his model. Figure 1 illustrates the relationships function (equation 2.4), and a pesticide cost between pest population and time in the model. function (equation 2.5). These are presented The economic threshold, as defined by Headley, below. is the level Pt-n to which the population should Pest population growth and kill functions: be reduced rather than the level population fPoert for to < ti Pt-_n- at which controls would be initiated. Previous definitions of the economic threshold (2.1) - by entomologists have emphasized this latter (Poeti K)erCt-ti) population level. The vast majority of pests do for ti < t < th not have a crucial stage when they are vulner- K = P(t,)] = K*(X, Poer") able to pesticides; hence, the question of when (2.2a) K*[X, to apply pesticides must be addressed. Indeed, (2.2b) K = K'(X, ti, Po, r)
1 This assumption is not made clear in the original paper where but is implicitly stated in a later letter from Headley to the r authors dated February 3, 1972: "In fact, the population =pest population growth, level Pt-n is entomologically determined as the crucial state to= planting time, of the insect" (italics ours). ti= pesticide application time, 200 / HALL AND NORGAARD Am. J. Agr. Econ.
th= harvest time, Product yield function: K= killed pests by pesticide application, Y = N - PO= initial pest population at to, (2.4) D(th) and where Y= at X= of physical yield harvest, quantity pesticides. to= 0, The kill that the of function indicates number and pests killed is a function of how much pesticide N = yield if no occurs from is applied and how many pests are present when physical damage the pesticide is applied. P(ti) is a function of ti, pests. Pesticide cost function: r, and Po; but, since the parameters Po and r are assumed to be fixed for any given planting (2.5) C = aX period and pest, the abbreviation is as follows: where (2.2c) K = K(X, ti). C= total cost of pesticides, Pest population damage function: X= number of units of pesticides, (2.3a) d(t) = bP(t) and a= cost of and a unit of rt2 purchasing applying (2.3b) D(2 - t) = d(t)dt pesticide. ti Profit can now be written: where (2.6a) ;r = Y - C = [N - Di- D2]- aX. = instantaneous rate of d(t) crop damage Substituting equation (2.3c) into (2.6a) re- in physical units due to pests (note sults in the following: that d(t) is piecewise continuous over to to th); - Ob T= 8N { (erth - eti) b=a parameter which specifies the r rate of in crop damage physical (2.6b) - units per pest; ? [Po -rtiK(X, ti)] and + Po(er"i - 1)} - aX. D(t2- i) =cumulative crop damage between The economic threshold is the population time ti and time t2. level P(ti) associated with the two decision variables, the optimum application time ti, and Total at harvest time can be broken damage the optimum quantity of pesticide X, which into two one before parts, pesticide application simultaneously maximize profits. Differenti- and one after: ating equation (2.6b) with respect to X and ,t r.ti tt rt h results in the first-order conditions: = 23cD(th - 10) d(1)dt + dt) dt (2.3c) J to (2.7a) 7rx=-a+b [e(t^-t)-1]Kx(X, ti)=0. - D1 + D2.
b where A (t) is the pest damage tolerance level, the maximum - = - D(th to) { (erh erti) pest level which, at each point in time, results r population in no discernible loss due to pest damage at harvest time. (2.3d) Two cases must be first is whether [Po - e-'tiK(X, ti)] investigated. The Po is less than A (to). The second is whether Poerti-K is less + Po(et' - 1)} than A(ti). For simplicity, suppose A is constant. If Po
[i= [1-e ('^-")]Kt,(X, t) Conclusions (2.7b) +rer(th-t)K(X, t) = 0. The foregoing model is an improvement on Headley's model in that both the timing and the and indicate that the Equations (2.7a) (2.7b) quantity of pesticide applied are variables. economic threshold is also a function of the Like Headley's model, it provides rigor to the time of the rate of harvest, pest population definition of the concept of economic threshold the rate of to growth, damage crops per pest, but is too simple for practical application. In the effectiveness of the the cost of pesticide, reality, the decision to spray is complicated by and the of the pesticide, price crop. Rewriting the presence of more than one pest and inter- and results in the equations (2.7a) (2.7b) relationships between pests, beneficial preda- following: tors, and parasites which may also be killed by ar the pesticide. Furthermore, the toxicity of some (2.7c) Kx(X, ti) = pesticides is persistent; weather, pest density, - vb[er(th-i) 1] and the availability of food, among other fac- rK(X, ti)er(th-t') tors, influence net population growth rates; (2.7d) Kt,(X, ti) are sometimes more and other times less er(th-ti) - 1 plants sensitive to pest damage; biological control Since th is greater than ti in both case 1, con- inputs can be introduced into the system; and the itself can be to reduce dition 2, and case 2, er(th-i)--1 is positive. system manipulated Therefore, since a, r, ,, and b are also positive, pest damage. Recent work by Carlson has Kx(X, ti) is positive, indicating that more brought out the importance of uncertainty and pesticide indeed kills more pests. Since K(X, ti) the risk preferences of pest managers [1]. As is positive for positive values of X, Kti(X, ti) is these additional factors are introduced, mathe- positive. This is intuitively appealing since, for matical models rapidly become unmanageable.4 later ti's, the pest population density increases, "Black box" approaches, which explicitly and a fixed quantity of pesticide would kill a recognize the uncertainty of future events and larger number of pests when the pest density is the risk preferences of the decision-maker, will greater. It is rather interesting to note that be necessary in the next generation of models. product and pesticide costs affect Kx but not K,i. This means if the of that, quantity pesti- negative quantities of pesticide. The second-order condi- cides to be applied is for some reason fixed, the tion irxx<0 and the previous result er(t^h-)-1>0 imply timing of application would not be affected by that Kxx References [1] CARLSON,GERALD A., "A Decision Theoretic Approach [3] National Research Council, Subcommittee on Insect to Crop Disease Prediction and Control," Am. J. Agr. Pests of the Committee on Plant and Animal Pests, Econ. 52:216-223, May 1970. "Insect Pest Management Control," Principles of Plant [2] HEADLEY, J. C., "Defining the Economic Threshold," and Animal Pest Control,Ch. 17, Vol. 3, 1969. presented at the National Academy of Sciences, Sym- [4] SAMUELSON,PAUL A., "Generalized Predator-Prey posium on Pest Control Strategies for the Future, Oscillations in Ecological and Economic Equilibrium," Washington, D. C., April 15, 1971 (in National Acad- Proceedings of the National Academy of Sciences, U.S.A. emy of Science, Pest Control Strategies for the Future, 68:980-983, May 1971. 1972, pp. 100-108).