**Analyzing Risk for IPM**

**(An incomplete draft)**

**October 16, 2012**

**Paul D. Mitchell**

Associate Professor

Agricultural and Applied Economics

University of Wisconsin, Madison, WI

, 608.265.6514

Basic IPM Model

This paper develops a basic IPM model to use for the analysis of risk management for IPM decision making. Model development begins with no risk or uncertainty, and then adds risk to the model, and finally adds risk preferences to the model. This basic model begins with the standard entomological of IPM as formulated by Pedigo (1996), but derives it by specifying farmer net returns from crop production so that the model can be more explicitly linked to the literature in the economics of risk management. Given this basic model, it is used in the next section to illustrate various concepts in risk management for IPM decision making.

Farmer Net Returns

A farmer plants a crop that yields output Y per unit area that sells in a competitive market at price P, so that the crop value per unit area is *V = PY. For example, Y is bu/ac of corn and P* is the corn price as $/bu and the crop value V = PY is $/ac. An insect pest causes crop injury resulting in proportional yield loss. With no pest control, per acre farmer returns are

(1)no = V(1 – no) –G,

where the subscript no denotes net returns and yield loss with no pest control and G is the per acre cost of production for all inputs other than pest control. With pest control, per acre farmer returns are

(2)t = V(1 – t) –C – G,

where the subscript t denotes net returns and yield loss with a pest control treatment and C is the cost of the pest control treatment.

Proportional yield loss depends on the amount of crop injury per pest (I), and the yield loss per unit of crop injury (D) and the pest population density (N). For example, crop injury in corn from corn rootworm larval feeding is measured by the node injury scale (Oleson et al. 2005), while the amount of stalk tunneling measures crop injury from second generation European corn borer larvae (Mitchell et al. 2002; Hurley et al. 2004; Hutchison et al. 2010). Thus, crop injury per pest (I) is the root rating due to one additional rootworm larvae or the centimeters of stalk tunneling by a second-generation European corn borer (ECB) larvae, and then loss per unit of crop injury (D) is the proportion of yield lost per root rating or per centimeter of stalk tunneling. Entomological models typically express this relationship multiplicatively as = IDN (Pedigo 1996), though as an empirical example in the next section demonstrates, this is not the only formulation possible.

Using the pest control treatment in some manner reduces the pest population density that causes damage, which is expressedmathematically as the proportion K that no longer causes damage. This proportion K is the efficacy of the pest control treatment, for example the proportion of the population that an insecticide kills. Thus, the pest population densitywithout treatment is N and when the pest control treatment is used, the population density isN(1 – K). Based on these densities, proportional yield loss with no pest control is no = IDN and when the pest control treatment is used, proportional yield loss is t = IDN(1 – K). Thus net returns as expressed in equations (1) and (2) become:

(3)no = V(1 – IDN) – G,

(4)t = V(1 – IDN(1 – K)) – C – G.

For this chapter, we focus solely on the case of a discrete pest control treatment (i.e., a treatment that is used or is not used) and will not examine cases in which the level of pest control is a continuous choice variable. For example, a farmer uses an insecticide at the recommended application rate, or does not use the insecticide; we do not examine cases in which the insecticide application rate is varied.

Net Returns and IPM

For IPM, the economic injury level (EIL) is the pest population density at which the cost of the pest control treatment equals the monetary loss from the pest population (Pedigo 1996, p. 248-249). In terms of the economic model expressed in equations (3) and (4), the EIL is the pest population density N at which t = no. Equating equations (3) and (4) and solving for N gives:

(5),

where the subscript nr indicates “no risk” in the sense that no variables are random. This EILnr is the standard entomological definition of the EIL(Pedigo 1996, p. 250). A pest population density exceeding this EIL implies that t exceeds no (i.e., treatment is not economical), while a pest population density below the EIL implies that no exceeds t (i.e., treatment is not economical). However, economic analysis of IPM goes beyond the derivation of the EIL and compares net returns under the available management systems to identify the most profitable system to recommend to farmers.

Examining equations (3) and (4) indicates that both are linear in the population density N, but with different slopes and intercepts. Specifically, equations (3) and (4) can be expressed as *no = V – G – VIDN and t = V – G – C – VID(1 – K)N*. Thus, net return in both cases are linear in the pest population density *N, with slopes of –VID and –VID(1 – K) respectively, where –VID < –VID(1 – K)as long as K > 0, so that t*decreases less steeply, and the intercept for *tis lower than for noby the amount C. Thus, no* at low pest densities exceeds *t, but at some density N = EIL, t exceeds no* and the farmer finds it economical to treat (see Figure 1). Graphically, the intersection of the two net returns lines defines the EILnr, the pest density at which it is profitable to switch from no pest control to using pest control.

A farmer using IPM first scouts first to determine the pest population density N and then makes the pest control decision. If N is less than the EIL, the farmer does not treat, and if N equals or exceeds the EIL, the farmer treats. Thus, in essence, for any given pest density, the farmer chooses whichever practice generates the highest net returns (no or t). However, the farmer does not actually earn this level of net returns, since scouting to determine the pest density is costly. Thus, net returns with IPM as a function of the observed pest density are:

(6)ipm = ,

where S is the scouting cost. Equation (6) implies thatat any given pest density *N, ipm is the maximum of no and t, minus the scouting cost S* (see Figure 1).

Figure 1 seems to indicate that IPM is not valuable—net returns with no pest control or with pest control are higher than with IPM at every pest density. However, this is true only if the farmer knows beforehand the pest population densityandcanthenchoose whether ornottotreat. Why would farmers spend money to scout if pest densities were already known? Figure 1 illustrates an important point for the economics of IPM—the pest population density must be variable or “risky” for IPM to be valuable. Without variability/risk, there is no need for IPM. However, pest densities often fluctuate between years or within a season,so that farmers do not know beforehand whether or not it is profitable to use a pest control treatment, and so scouting to determine the pest density to makea control decisioncan have value. This variability in pest densities leads to variability in net returns, and thus to risk as economics defines it, but this variability is not the only source of risk when using IPM.

Most variables in equations (3), (4), and (6) are likely random. For example, crop yields and prices are usually not certain, but depend on weather and markets outside of farmer control, so that the crop value V is random. Similarly, proportional yield loss per pest as determined by I and Dand control efficacy as determined by Koften depend on environmental factors such as weather, and so are also random. Finally, costs G, C, and S can also be random due to unanticipated input price changes or problems with labor or product availability. As a result, we next add risk to the basic model.

**IPM Model with Risk**

We add risk by assuming all variables in this IPM model (V, I, D, K, N, C, G, and S) are random with known statistical distributions. To keep the analysis general, we assume each is an independent random variable with mean *i and standard deviation i, where i*{V, I, D, K, N, C, G, S} indexes the variable. Because net returns are now random, expected (average) net returns measures the profitability of each pest control system, i.e., the probability weighted average of net returns over all possible outcomes with the probability weights implied by the statistical distributions. In terms of standard risk vocabulary, this case assumes the farmer is risk neutral, i.e., maximizes expected net returns and does not respond to changes in the variance of net returns. Finally, assuming the random variables are independent is important, since the expected value of the product of two independent random variables is the product of their means, while without independence, determining the expected value is more difficult (Casella and Berger 2002, p. 139-168).

With these assumptions, expected net returns with no pest control and with the pest control treatment are

(7),

(8).

Due to two crucial assumptions, these equations are essentially the same as equations (3) and (4), except that the known certain values of each variable are replaced with their respective means. The crucial assumptions are first, that each random variable enters the net returns specification linearly and second, that all random variables are independent of the other random variables. Relaxing these assumptions implies more complex expressions and less intuitive results that are explored in the next section.

Graphically, equations (7) and (8) are illustrated in the top plot in Figure 2 under the simple assumption that the random pest density has only two levels, low and high, each with probability ½. Randomness in the other variables does not change the plot, except to replace the other variables with their means. Following the rules of expected values, expected net returns for both systems is ½ the net return with the low pest density plus ½ the net return with the high pest density. Because net returns are linear in the pest density, expected net returns are on the net returns lines where it intersects the expected pest density N. In other words, expected net returns with a random pest density are the same as net returns at the expected pest density, implying that it does not matter when the expected value is taken, before the pest density enters the net returns function or after. This result depends crucially on the linearity and independence assumptions. Assuming a more realistic statistical distribution other than two pest densities each with probability ½ would not change this result. Finally, Figure 2 shows E[t] exceeding E[no], but this is not a general result, but depends on how the expected pest density compares to the EILrn (defined below). If the EILrnexceeds *N, then E[t] will exceedE[no], but if Nexceeds EILrn, then E[no] will exceed E[t*]. Again, assuming a more realistic statistical distribution other than two pest densities each with probability ½ will not change this result;rather it depends on the linearity and independence assumptions.

Equations (7) and (8) are the appropriate measures for comparing the two pest management systems beforehand to choose the system that will on average be most profitable—never using pest control or always using pest control. However, to derive the EIL and expected returns with IPM, these expressions are not appropriate, as they are the unconditional expected net returns. Expected net returns conditional on the observed pest density N are needed instead. In other words, given an observed pest densityN, what are expected net returns if no pest control is used and if the pest control treatment is used?

With the linearity and independence assumptions, expected net returns conditional on the observed pest density are fairly intuitive. Specifically, expected net returns conditional on the observed pest density N are:

(9),

(10).

These expressions are the same as equations (7) and (8) for the unconditional expected net returns, except that the observed random pest density N replaces the mean pest density N. Graphically, these expressions are the lines plotted in Figure 2. The EIL for IPMis derived by equating equations (9) and (10) and solving for N, which gives

(11),

where the subscript rn denotes risk neutral in the sense that this EIL is the EIL for a risk neutral farmer. Notice that this EILformula is the same as the EIL with no risk as reported in equation (5), except that the known certain values of each variable are replaced with their respective means. Again, this result depends crucially on the linearity and independence assumptions. Figure 2graphically illustrates the derivation of this EILrn as the intersection of expected net returns conditional on the observed pest density with no pest control and with a pest control treatment.

When using IPM, farmer expected net returns will be a weighted average of the expected conditional expected net returns when no pest control is used and when the pest control treatment is used, minus scouting costs, where the weights for the average are determined by how likely the pest density is above or below EILrn. Mathematically, the“expected conditional expected net returns” is the expected value over the pest density N of the conditional expected net returns, or EN[E[no|N]] and EN[E[t|N]], where EN[∙] is the expected value of the term in square brackets over the random variable N. Given the linearity and independence assumptions, EN[E[*no|N]] = E[no] and EN[E[t|N]] = E[t*],but this simplification is not a general result. Combining these results, expected net returns when using IPM are

(12)E[*ipm] = Pr[N < EILrn]EN[E[no|N]] + Pr[N ≥ EILrn]ENE[t|N]] – S*

E[*ipm] = Pr[N < EILrn]E[no] + Pr[N ≥ EILrn]E[t] – S*,

where Pr[∙] is the probability that the event in the square brackets occurs and the simpler second expression follows due to the linearity and independence assumptions, and so the resulting simplification is not a general result.

The probability that N exceeds or falls below EILrn will depend not only on how the EIL relates to N, but the complete statistical distribution of the pest density N, including the spread (variance) of the pest density around its mean as well as its skewness. This probability and its dependence on the qualities of the statistical distribution is captured by F(N), the cumulative distribution function for the statistical distribution of the pest density N. Following the rules of probability theory, the probability that N is less than the EILrn is Pr[N < EILrn] = F(EILrn) while the probability that Nequals or exceeds the EILrn is Pr[N ≥ EILrn] = 1 – F(EILrn). Combining this result with previous results, expected net returns when using IPM can also be expressed as

(13)E[*ipm] = F(EILrn)E[no] + (1 – F(EILrn))E[t] – S*.

Figure 2 illustrates the derivation of the unconditional expected net returns under the simple assumption that the pest density has two levels (low and high) each with probability ½, so that Pr[*N < EILrn] = F(EILrn) = Pr[N ≥ EILrn] = 1 – F*(EILrn) = ½. Graphically, E[ipm] is where the line drawn between E[*ipm |N = Nlow] and E[ipm |N = Nhigh*] intersects the expected pest density N. The graphical simplicity of this result depends on the assumption of two levels for the pest population, but the result serves to illustrate the point of how to derive expected returns when using IPM. This graphical simplicity is lost if the pest population density follows a more realistic statistical distribution, in which case equation (13) can be used. However, regardless of the distribution of the pest population density, with equation (13), expected net returns with IPM is a linear combination of expected returns with no pest control (E[no]) and expected returns with pest control (E[t]), weighted by the probability that the pest population density is below and above the threshold (1 – F(EILrn) and F(EILrn)), minus the cost of scouting (S). More specifically, expected returns with IPM requires integrating the dotted line in the bottom plot in Figure 2 over the pest population density N.

**Points to Mention**

1)How variability in pest population increases value of IPM because are integrating over a convex function

2)If relax independence, big mess

3)If have non-linearity: not hard to plot graphically, mathematics gets more complicated with random variables

4)Can show how parameter changes affect EIL and profit

**IPM Model with Risk and Non-Linear Losses**

**IPM Model with Risk, Non-Linear Losses, and Grower Risk Preferences**

Figure 1. Net returns plotted versus the pest population density with no pest control and with a pest control treatment (top) to show derivation of the EIL with no risk (EILnr) and net returns plotted versus the pest population density (bottom)to show derivation of net returns with IPM.