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Parsimonious Multi-Dimensional Impact Assessment*

John M. Antle Agricultural and Resource Economics

Oregon State University

June 2011 Abstract This article develops the conceptual and empirical foundations for a parsimonious, generic modeling approach to multi-dimensional (i.e., economic, environmental and social) impact assessments of agricultural technologies and environmental change. Joint distributions between technology adoption and outcome variables are characterized, and used to analyze the selection effects of adoption on a general class of impact indicators. The approach is implemented with a generic model that can be parameterized with low-order moments of outcome variables. A case study of adoption of a high-yielding maize variety in Kenya illustrates the model’s use and confirms theoretical results.

Key Words: adoption, economic, environmental, impact, assessment, multi-dimensional, social, parsimony. *Antle, J.M. 2011. “Parsimonious Multi-Dimensional Impact Assessment.” American Journal of Agricultural Economics, in press.

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Parsimonious Multi-Dimensional Impact Assessment

The objective of this article is to develop the conceptual and empirical foundations for a

parsimonious, generic modeling approach to multi-dimensional (i.e., economic,

environmental and social) impact assessment of agricultural technologies. This approach

is designed to respond to the growing demand for assessments that anticipate possible

impacts of new technologies and the effects of environmental change (e.g., Adams et al.

2004; Organization for Cooperation and Development 2006; Bill and Melinda Gates

Foundation 2009; Integrated Pest Management Collaborative Research Support Program

2009; International Food Policy Research Institute 2009; National Institute for Food and

Agriculture 2010).1 The demand for multi-dimensional impact assessments poses a

substantial challenge to the research community. “In terms of both budgetary support and

human capital, a disaggregated multi-dimensional impact study can be quite demanding

and costly….The supply of these studies is more likely to be constrained by lack of

funding than the other types….” (Walker et al., 2008, p. 7). Nevertheless: “The

desirability of moving…along the impact pathway is unquestioned. As donors want to

see ever more comprehensive impact assessments, so ways have to be found to

accommodate their wishes… even when resources for carrying out these…studies are not

forthcoming.” (Walker et al., 2008, p. 14).

There is a mature literature on technology adoption and its economic impacts, an

emerging literature on adaptation to environmental change, and extensive literatures on

environmental and social impact assessment beyond the boundaries of economics.

Economists have evaluated factors affecting the extent and rate of adoption of various

agricultural technologies, and assessed their economic impacts (for reviews, see Feder,

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Just and Zilberman 1985; Alston, Norton and Pardey 1995; Sunding and Zilberman 2001;

Walker et al. 2008; Swinton and Norton 2009; Foster and Rosezweig 2010). However,

the econometric methods developed to model adoption and estimate economic or other

impacts are useful only for ex post analysis, and only when suitable cross-sectional or

panel data sets are available (e.g.,see the review by de Janvry, Dustan and Sadoulet

2010). Even then, observation or prediction of key variables in impact assessment, such

as the adoption rate, is problematic.2

A complementary but distinct literature has analyzed environmental – and in

some cases health, nutritional and other social – impacts and tradeoffs associated with

agricultural technologies and policies, e.g. Crissman, Antle and Capalbo (1998); Lee and

Barrett (2001); Khanna, Isik and Zilberman (2002); Moyo et al. (2007); and Xabadi,

Goetz and Zilberman (2008). Recognition of the importance of physical and economic

heterogeneity in agricultural systems has led to the development of increasingly complex

models that represent bio-physical and economic processes and their interactions at

various scales, including the household or local community (Taylor and Adelman 2003;

Holden 2005), as well as regional and global scales (van Ittersum et al. 2008; Rosegrant

et al. 2008). However, models based on a “representative farm” construct cannot predict

adoption rates or estimate distributional impacts of adoption; models that do usefully

represent heterogeneity are typically data-intensive, and often involve a coupled system

of complex, spatially-explicit disciplinary models. While these models are useful research

tools, they are typically too costly and data-intensive to be used for impact assessment

beyond the case studies for which they were developed.

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In this article I propose a parsimonious, generic simulation model for multi-

dimensional impact assessment of technology adoption and its consequences in

heterogeneous populations. In the first part of this article I develop the conceptual

foundations for this model, by analyzing the implications of economically rational

technology adoption behavior for the properties of economic, environmental and social

outcome distributions and impact indicators based on them. In this model, as in most

actual situations, adoption is usually less than 100 percent, and farmers select themselves

into adopting and non-adopting sub-populations. I show that the resulting distributions of

outcomes are truncated according to the adoption process, and that the effects of selection

on impact indicators depend on the properties of the joint distributions of variables

influencing adoption (e.g., expected returns) and outcomes (e.g., income, water

contamination, nutrition). A key implication is that parameters of these joint distributions

(i.e., correlations) are needed to obtain accurate estimates of impact in the sub-

populations of adopters of a new technology. However, I also show that under certain

conditions, the aggregate relationship between adoption and impact can be estimated

accurately without knowing the joint distributions of adoption and outcome variables.

These conditions are approximately true in the case study presented, and if found to be

valid more generally, it should be possible to estimate aggregate impacts accurately even

when adoption and outcome data are collected independently.

The second part of this article uses these results to construct a parsimonious,

generic impact assessment model that can be parameterized with the low-order moments

of outcome distributions. The generic model structure facilitates assessment design and

lowers data costs by helping researchers to identify essential data, avoid collection of

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superfluous data, and integrate data collection into technology development projects. The

generic structure and parameter parsimony also facilitate the use of the various types of

data that may be available, including farm survey, experimental, modeled, and

aggregated secondary data, as well as expert knowledge. The parsimonious structure also

provides a framework in which sensitivity analysis of model parameters can be

conveniently carried out. A case study illustrates the use of this model, and confirms the

theoretical properties of output distributions. In the concluding section I summarize and

discuss extensions.

Adoption, Selection and Impact in Heterogeneous Populations

My goal in this section is to characterize the properties of economic, environmental and

social outcome distributions associated with technology adoption, and analyze their

implications for impact assessment. Here I introduce the theoretical model and present

results, with technical details and proofs in the Appendix. In this analysis I use a

threshold model of adoption in which farms are presented with the opportunity to

continue operating with the current production system, system 1, or switch to an

alternative system 2. The analysis is comparative static and I abstract from the time

dimension of adoption to simplify the analysis.

Farms are assumed to choose a system to maximize a function v(h) where h = 1,2

indexes the production system and all attributes associated with it, including prices and

other factors affecting adoption, such as geography and infrastructure. I interpret v(h) as

expected returns, but it can be defined more generally as a certainty equivalent, or as

some other objective function that depends on the characteristics of the farms and the

system being used, such as farm size or land quality. This objective function induces an

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ordering ω ≡ v(1) – v(2) over all farms, such that for the adoption threshold a, ω > a for

those farms using system 1 and ω < a for those using system 2. The adoption variable ω

is spatially distributed across the landscape according to the density ϕ(ω), which is

generally a function of prices and other exogenous variables, taken to be implicit in the

function for notational convenience. The proportion of farms using system 2, referred to

as the adoption rate of system 2, is given by the cumulative distribution function

(1) r(2,a) ≡ ∫ φ𝑎𝑎−∞ (ω) dω, 0 ≤ r(2,a) ≤ 1,

and the share of farms using system 1 is r(1,a) ≡ 1 – r(2,a). For convenience, I assume

that the adoption ceiling is 100 percent. I refer to r(1,a) and r(2,a) as adoption “rates,”

recognizing that we would distinguish the share of adopters at a point in time from its rate

of change in a dynamic analysis.

An important feature of this model is that it allows a “technology” to be

represented realistically as a set of management practices distinguished by the use of

certain technological components, but all farms need not be using the technology in

precisely the same manner. Thus, in this model, the only distinguishing feature of each

system is that it gives rise to different expected returns for producers. This feature is

important, because even for a well defined technology such as a seed variety, a wide

array of management practices is typically applied (also see Swinton and Norton, 2009).

This simple yet general adoption model has the following important implications

for adoption behavior:

Proposition 1. For ω ~ (µω , 𝜎𝜎𝜔𝜔2 ) and ω ∈ [𝜔𝜔 ,𝜔𝜔�]: (i) 𝜎𝜎𝜔𝜔2 > 0, µω ∈ (𝜔𝜔 ,𝜔𝜔�) and a ∈ (𝜔𝜔

,𝜔𝜔�) imply 0 < r(2,a) < 1; (ii) µω < a and 𝜎𝜎𝜔𝜔2→ 0 imply r(2,a) → 1; (iii) µω > a and 𝜎𝜎𝜔𝜔2→

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0 imply r(2,a)→ 0; (iv) if the distribution is symmetric, µω < a implies 0.5 < r(2,a) < 1,

µω > a implies 0 < r(2,a) < 0.5, and µω = a implies r(2,a) = 0.5.

The proof offered here is heuristic. For part (i), observe that for a non-degenerate

distribution of ω with an adoption threshold within the support of the distribution, there is

some mass above and below a, and thus the adoption rate is positive but less than 100

percent. For parts (ii) and (iii), as σω2 approaches zero, the distribution concentrates at the

mean, so that if µω < a all farms adopt, and if µω > a none adopt. If the distribution is

symmetric, µω < a means that more than half the density is below a so more than 50

percent adopt, and conversely less than 50 percent adopt if µω > a.

Proposition 1 tells us that the economic behavior of farmers is likely to result in a

level of adoption between 0 and 100 percent, except in the extraordinary situation where

one system dominates the other for almost every farm and the variance of opportunity

cost is close to zero. An important implication of this adoption model is that incomplete

adoption can be caused by heterogeneity in the conditions in which a system is operating,

such as heterogeneity in soils, climate, prices, transportation costs, and the farm

household’s characteristics (Antle et al. 2005). Incomplete adoption also can be caused

by constraints on adoption such as risk aversion and access to information (for a review

of this literature see Sunding and Zilberman 2001; for further critical discussion see Suri

2011). In the spirit of the parsimony principle, the adoption model based on heterogeneity

in economic returns is proposed as an appropriate starting point for analysis of adoption

and impact assessment. If this model fails to predict adoption well enough for the

purposes of impact assessment, then it may be worthwhile to consider more elaborate

models. For example, the opportunity cost concept could be generalized to include

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farmers’ “willingness to adopt” a new system, with ω re-defined as the difference in

expected returns plus an “adoption premium” (which could be either positive or negative)

reflecting attitudinal (e.g., risk aversion) or other factors causing decision makers to

deviate from the opportunity cost decision rule. Using the logic of Proposition 1, it can be

shown that if the adoption premium is randomly distributed in the population with a non-

zero mean and a positive variance, the effect on the adoption rate depends on the sign of

the mean adoption premium and whether it is greater or less than the adoption threshold.

In addition to economic outcomes v(h), I consider an environmental or social

outcome z(h). The marginal distributions for v and z, given h, are defined as φ(k | h) for k

= v,z. The adoption variable ω and the outcomes k = v, z are influenced by many of the

same factors, and thus are jointly distributed. To construct these joint distributions, we

can define the conditional outcome distributions φ(k | ω,h) for outcome k, given ω, for

system h, and then use Bayes’ rule to obtain φ(ω,k | h) = φ(k | ω,h) ϕ(ω). In the Appendix,

I use this fact to show that the distribution for the sub-population using each system is the

joint outcome distribution between ω and k = v, z, truncated according to ω > a for

system 1 and ω < a for system 2:

(2) φ(ω,k | h,a) ≡ φ(ω,k | h)/r(h,a) = φ(k | ω,h) ϕ(ω)/r(h,a).

Note that φ(ω,k | h,a) indicates a distribution truncated from below by a for system 1, and

truncated from above by a for system 2, whereas φ(ω,k | h) is a distribution defined over

the entire population. The joint distribution of ω and k = v,z in a population using both

systems is a mixture of the distributions defined in (2), with mixing proportions r(h,a)

(McLachlan and Peel 2000). I also show in the Appendix (equation A3) that integrating

φ(ω,k | h,a) over the interval ω > a for system 1 and over ω < a for system 2 gives the

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outcome distributions χ(k | h,a) for outcome k, conditional on the adoption threshold a.

This fact links the adoption process to the outcome distributions conditional on adoption,

and thus plays a key role in the analysis of the properties of impact indicators. An

outcome distribution for the entire population composed of non-adopters and adopters is

thus a mixture of the distributions of each group, given threshold a:

(3) χ(k | a) = r(1,a) χ(k | 1,a) + r(2,a) χ(k | 2,a).

Figure 1 illustrates the initial marginal outcome distribution for system 1 before

system 2 is introduced, φ(k | 1), the outcome distributions for non-adopters and adopters

conditional on adoption, χ(k | h,a), h =1,2, and the aggregate outcome distribution

(equation 3). In this example, before system 2 is introduced, 40 percent of the farms have

an outcome k that exceeds the outcome threshold τ. For example, k could represent a

nutritional outcome such as nutrient consumption per family member, and τ could

represent the minimum daily requirement for good nutrition. After system 2 is

introduced, some farms adopt system 2 while others continue to use system 1. The

characteristics of the farms selecting themselves into the adopter and non-adopter groups

give rise to distinct outcome distributions for each group. In this example, if system 2

involves production of a crop with more nutrients and adopting farms are producing for

their own household consumption, then farms adopting system 2 would have a

distribution for k with a higher mean than the initial marginal distribution associated with

system 1; conversely, non-adopters would have a lower mean. In the example, 85 percent

of adopter farms exceed the outcome threshold τ, whereas only 30 percent of non-

adopters do. When the two sub-populations are combined, 60 percent exceed the outcome

threshold.

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Figure 2 provides a graphical interpretation of the properties of the joint

distribution φ(ω,k | h,a) that gives rise to the distributions of k for the adopter and non-

adopter groups illustrated in figure 1. In the figure, the dashed ellipsoid is a contour of

equal density for the joint distribution between opportunity cost (ω) and nutrient

consumption (k) for system 1; the solid ellipsoid is the contour of equal density for

system 2. With adoption threshold a = 0, the outcomes for farms that continue to use

system 1 are represented by the positive half-plane, and the outcomes for adopters of

system 2 are represented by the negative half-plane. The figure shows the case of a

negative correlation between k and opportunity cost, reflecting the situation described in

the previous paragraph, and represented in figure 1, where adopters improve nutrition

when they adopt system 2. Note that the mean of the outcome distribution, µk(2,0), is

obtained by integrating over k for ω < 0, and is higher for the adopter group than for the

overall population, as in figure 1. Indicators defined relative to the outcome threshold τ

are represented by the areas identified in figure 2, as explained in the figure caption. Note

that if the adoption threshold a were different than zero, the mean and threshold

outcomes would change systematically. The analysis and example presented below

explore these relationships.

As the above example suggests, economic, environmental and social indicators

can be any function of the outcome distributions for each system. In this presentation, the

indicators are defined as:

(4) Ik(h,a,τ) ≡ ∫ ι𝛕𝛕 (k) χ(k | h,a) dk, k = v,z,

where ι(k) is a function of k, and τ defines a threshold value for the variable. For

example, if k is per-capita income, setting ι(k) = k and defining τ to include all feasible

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values of k, then Ιk(h,a,τ) is mean per-capita income for the farms using system h.

Ιk(h,a,τ) also can represent a general class of threshold indicators. For example, if k is

per-capita income, then setting ι(k) = 1 and defining τ as the poverty line, Ιk(h,a,τ)

becomes the headcount poverty index; letting ι(k) = (τ – k)/τ produces the “poverty gap”

relative to the poverty line τ, as defined by Foster, Greer and Thorbecke (1984).

Measures of vulnerability to exogenous environmental changes, such as climate change,

can be represented using similar indicators (Antle et al. 2004).

Using (3) and (4), indicators for the entire population of adopters and non-

adopters are:

Ιk(a,τ) ≡ ∫ ιτ (k) χ(k |a) dk

(5) = r(1,a)∫ ι(𝑘𝑘)χτ (k | 1,a) dk + r(2,a)∫ ι(𝑘𝑘)χτ (k | 2,a) dk

= r(1,a) Ιk(1,a,τ) + r(2,a) Ιk(2,a,τ), k = v, z.

Equation (5) demonstrates that this class of indicators exhibits the additive

decomposability property of the FGT-type indicators, meaning that the same type of

indicator can be used for sub-populations and for the entire population. Equations (2) –

(5) also show that impact assessment generally requires knowledge of the adoption

process, as embodied in ϕ(ω), and the conditional distributions of outcome variables, χ(k

| h,a), for each system. The properties of these distributions have important implications

for impact assessment, summarized below, and proved in the Appendix. The expectation

of ι(k) for system h, taken with respect to φ(k | ω,h), at the point ω = a, is written as

Mk(a,h,τ) ≡ ∫ ι𝛕𝛕 (k) φ(k | a,h) dk. This expectation plays an important role in the analysis,

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as demonstrated in the Appendix, because it represents the marginal effect of the

threshold a on the indicator (4).

Proposition 2. dIk(h,a,τ)/dr(2,a) ><

0 as θk(h) ><

0 where θk(h) is the correlation of ω with

k = v(h),z(h), with the signs on θk(h) reversed when τ is an upper threshold.

Proposition 3. If dΙv(a,τ)/dr(2,a) >( <) 0 for a < 0 and 𝜕𝜕𝑀𝑀𝑣𝑣(𝑎𝑎 ,2,𝜏𝜏)𝜕𝜕𝑎𝑎

< ( >) 𝜕𝜕𝑀𝑀𝑣𝑣(𝑎𝑎 ,1,𝜏𝜏)𝜕𝜕𝑎𝑎

, then

Iv(a,τ) is a concave (convex) function of a with a unique maximum (minimum) at a = 0.

Corollary 1. If expected returns v(h) are normally distributed, then mean and aggregate

returns in the farm population are maximized at a = 0.

Corollary 2. If 𝜕𝜕𝑀𝑀𝑘𝑘 (𝑎𝑎 ,2,τ)𝜕𝜕𝑎𝑎

= 𝜕𝜕𝑀𝑀𝑘𝑘 (𝑎𝑎 ,1,τ)𝜕𝜕𝑎𝑎

, then Ik(a,τ) is linearly related to the adoption rate

for system 2, with the slope of this relationship determined by the marginal outcome

distributions such that dIk(a,τ)/dr(2,a) = Ik(2,+∞,τ) – Ik(1, –∞,τ).

Corollary 3. If ω and k = v(h),z(h) are jointly normally distributed, and if the covariance

of ω and k(1) is equal to the covariance of ω and k(2), then Corollary 2 is satisfied for

indicators defined as population means.

Corollary 4. A sufficient condition for Corollary 2 to be satisfied is ∂φ(k|a,1)/∂a =

∂φ(k|a,2)/∂a.

Proposition 2 demonstrates that impacts in the sub-populations of adopters and

non-adopters vary systematically with the adoption threshold a, and thus vary with the

adoption rate r(2,a), depending on the correlation in the population between the adoption

variable ω and the outcome variable. Observe that indicators for both the adopting and

non-adopting sub-populations change as farms switch from the non-adopter to the

adopter group. This means that estimates of impacts in each group that are made treating

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adoption and outcomes as statistically independent will be biased, with the direction of

bias depending on the sign of the correlation between adoption and outcome variables.

Proposition 3 provides sufficient conditions for an economic indicator to be non-

linearly related to a, and to attain a maximum (or in the case of an indicator like a poverty

rate, a minimum) at the adoption threshold a = 0 where each farm chooses the system

with the highest expected value. This result is important because it shows, in combination

with Proposition 1, that economically rational technology choices lead to an

economically efficient adoption rate that maximizes the value of positive economic

outcomes. An important implication is that constraints on adoption, as well as actions or

policies that force farms to adopt an unprofitable system, will result in lower aggregate

economic returns than the economically optimal adoption rate. Corollary 1 demonstrates

that mean and aggregate returns are maximized at a = 0 in normal populations.

Corollary 2 shows that when the conditional means of the outcome distributions

of the two systems are affected equally by a, the impact indicator in the full population is

linearly related to the adoption rate, and the impacts can be calculated using the marginal

outcome distributions for each system. Corollary 3 shows that when outcome

distributions are normally distributed, as in the model presented below, Corollary 2 holds

for mean indicators when the covariances between opportunity cost and the outcome

variable are equal for the two systems. While this condition sounds restrictive, it is a

plausible condition in important cases, as illustrated by the example presented below.

Corollary 4 provides a sufficient condition for Corollary 2 for all indicators, and implies,

for example, that Corollary 2 is satisfied when the outcome distributions differ only by

their means.

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A Parsimonious Model for Impact Assessment

The previous section shows that the impacts of adoption can be quantified using the joint,

marginal and conditional outcome distributions defined for populations using system 1 or

system 2. In this section, I construct a parametric model of these distributions and discuss

its implementation. The following definitions will be used for parameters of the joint and

marginal distributions for all farms, where outcomes are indexed by k = v,z and systems

are indexed by h = 1,2, and k(h) refers to outcome k for system h:

µk(h) ≡ mean of k(h)

σ𝒌𝒌𝟐𝟐(h) ≡ variance of k(h)

σ𝛚𝛚𝟐𝟐 ≡ variance of ω

ρk ≡ correlation between outcomes k(1) and k(2)

κk(h) ≡ correlation between outcomes v(h) and k(h)

θk(h) ≡ correlation between outcome k(h) and ω.

Three correlations play a role in the model: ρk represents between system correlations of a

given outcome k; κk(h) represents within-system correlations between economic returns v

and outcome k; and θk(h) is the correlation between outcome k(h) and opportunity cost.

Opportunity cost is assumed to be normally distributed:

(6) ω = µv(1) – µv(2) + σωε, ε ~ N(0,1)

(7) σω2 = σ𝑣𝑣2(1) + σ𝑣𝑣2(2) – 2ρ𝑣𝑣 σ𝑣𝑣(1) σ𝑣𝑣(2).

Normality is not an essential assumption, but it is analytically convenient and appropriate

for a parsimonious model because the normal distribution is itself parsimonious. Note

that v(h) represents expected returns, so non-normally distributed realized returns are

consistent with this model. Antle and Valdivia (2006) and Antle et al. (2010) present

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validation of this model for analysis of ecosystem services supply by comparing it to

more elaborate models. To utilize the results presented in the previous section, I will also

assume that the environmental or social outcomes represented by z are normally

distributed. Normality is a particularly useful assumption for parameterization of the

truncated distributions discussed in the previous section, both for its parsimony and for

the well-known, tractable properties of the moments of truncated normal distributions. If

outcome distributions are non-normal, stratification of a population, e.g., by farm size,

can often improve the normal approximation.

Using the above definitions, the correlation between k(h) and ω = v(1) – v(2) is:

(8) θk(h) = {σv(1) κk(1) ρkh-1

– σv(2) κk(2)ρk2-h}/σω.

In the Appendix, I show that the means and variances of the distributions of the outcome

variables, for the sub-populations of farms using system h, with adoption threshold a, are:

(9) µk(h,a) = µk(h) – (–1)h σk(h) θk(h)λ(a,h),

(10) σ𝒌𝒌𝟐𝟐(h,a) = σ𝒌𝒌𝟐𝟐(h) {(1 – θk(h)2) + θk(h)2 (1 + (–1)h-1aλ(a,h) – λ(a,h)2)},

where λ(a,h) is the inverse Mills’ ratio associated with the truncated opportunity cost

distribution of system h. Using equations (9) and (10) one can verify the propositions

presented in the previous section.

Summarizing, we can see that the model outlined above involves 5 parameters of

the distribution of ω, the means and variances of v(1) and v(2) and their correlation. In the

case of outcome variables based on v, no additional parameters are required, as noted

above. For each non-economic outcome variable, there are seven additional parameters, a

mean and a variance for each system, and the three correlations defined in equation (8).

Thus, with m non-economic indicators, the total number of parameters is equal to 5 + 7m.

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This relatively small number of parameters makes this model easy to interpret and

convenient for analysis of parameter uncertainty.

System Design, Data, and Parameter Estimation

The first step in model implementation is definition of the population for the analysis,

including any appropriate stratifications, e.g., by geographic or socio-economic criteria.

The second step is the description of the systems being modeled and identification of the

impacts to be quantified. The third step is parameter estimation. The ideal data would be

a statistically representative sample from the population being modeled, with pair-wise

observations of both systems, to allow for estimation of the means and variances for each

system as well as correlations within and between systems. Although such ideal data are

not likely to be available, this concept is useful to guide the choice of data.

Although an extensive discussion of parameter estimation methods is beyond the

scope of this article, some basic observations illustrate some of the possible techniques.

In many cases, sample survey data will be available to represent system 1 and its impacts,

but will not be available to estimate the parameters of system 2. In such cases, the

challenge is to obtain the best data available to approximate the population parameters.

For example, in many analyses of agricultural technology impacts, only one component

of a multi-component system is directly impacted by the technology, as in the example

presented below of a high-yielding maize variety. In this case, the data for system 1

obtained from a survey are also relevant to system 2, with minor modifications, based on

knowledge of the technology and practices farmers are utilizing. In many cases,

experimental data, or data obtained from bio-physical simulation models, can be used to

estimate changes in crop yields and environmental impacts of changes in land

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management. Changes in cost of production can be estimated from expected changes in

input use, although care must be exercised when higher labor costs are involved, because

experimental data and even on-farm trials typically provide limited information about

changes in labor requirements and the variability in the population.

A number of methods can be used to characterize the variability of system returns

and between-system correlations (Antle and Valdivia 2006; Antle et al. 2010). It is often

possible to use knowledge about system 1 and system 2 to put bounds on parameters

when they cannot be measured with statistically representative data. For example, when

system 2 involves a modification of some components of system 1, but other components

are unchanged, we can put bounds on the variability of system 2. Even when survey data

are available representing each system, observations are not usually matched for the

calculation of between-system correlations, so one solution is to use propensity score

matching. Moreover, we can infer that the correlation ρv between the two systems is

generally positive, and will be closer to 1 the more similar the two systems are, so

reasonable bounds can often be placed on this parameter. Variance components analysis

also can be used to approximate variances and correlations by combining observations of

system 1 with experimental or a priori information about system 2.

Another parsimonious feature of this model is its use of low-order moments of

outcome distributions. These parameters can be estimated reliably with small samples

using standard “method of moments” estimates of means, variances and correlations and

their standard errors. This feature contrasts with structural or reduced-form econometric

models that typically rely on large sample properties of estimators, and must contend

with endogeneity and identification issues. Since most survey data are cross-sectional, in

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econometric production models multicollinearity is typically severe and statistical

reliability of parameter estimates is correspondingly low.

The analysis presented above shows that the correlations defined in equation (8)

play an important role in impact assessment. For an indicator based on economic returns

v, noting that κv(1) = κv(2) = 1 in equation (8) gives

(11) θv(h) = {σv(1)ρvh-1 – σv(2)ρv

2-h}/σω.

This equation in turn implies that θv(2) < θv(1), and following the analysis presented in

the Appendix, Proposition 3 shows that the relationship between an aggregate indicator

based on v and the adoption rate of system 2 will be non-linear with a maximum at a = 0.

Since θv(h) depends on the parameters that are used to estimate the adoption rate (see 6

and 7), no additional data are required for its estimation.

For non-economic indicators, data are needed to estimate the within-system

correlations κk(h) and the between-system correlations ρk in equation (8). Since these

correlations are not generally available in the literature, they will need to be obtained

from field measurements or from modeling studies, at least until a better understanding of

these parameters is established. For environmental outcomes, many process-based

models are available that can be used to simulate outcomes associated with each system,

as in the example presented below. In the case of social outcomes, models are not

typically available, so when technology adoption can be observed, survey data can be

used. In cases where survey data are not available, or when system 2 cannot be observed,

we can often infer the signs of κk(h) and ρk based on knowledge of the processes. Further,

ρk is likely to be positive and relatively close to 1 unless the processes generating

19

outcomes in the two systems are substantially different, as the example presented below

illustrates (see Table 1).

Uncertainty, Scenarios and Sensitivity Analysis

An important feature of most impact assessments is uncertainty, including model

uncertainty (e.g., the parametric form of the outcome distributions), parameter

uncertainty, and scenario uncertainty. The latter refers to the factors assumed to be

motivating adoption (e.g., the assumption of adoption based on expected returns),

exogenous economic variables such as prices and non-agricultural income, and

demographic characteristics of the relevant populations. Model uncertainty involves the

form of the opportunity cost and outcome distributions.

The model outlined above provides a useful framework for investigation of

parameter uncertainty using sensitivity analysis, by varying one or several parameters

over plausible ranges. The parsimonious representation helps to address the

dimensionality problem implicit in such analyses. As emphasized elsewhere (Walker et

al. 2008), a key factor in a technology impact assessment is the adoption rate. As the

example presented below illustrates, this model is designed to assess the sensitivity of

impact indicators to the adoption rate by varying the adoption threshold parameter a.

When other parameters of the model are estimated statistically, Monte Carlo simulation

can be used to investigate the effects of parameter uncertainty on estimates of adoption

and impact. This model also provides a useful tool to assess the importance of model

parameters that cannot be estimated with available data, providing a way for researchers

to assess the benefits and costs of collecting additional data.

20

Adoption and Impact of a High-Yielding Maize Variety in Machakos, Kenya

I now illustrate the use of this approach with an impact assessment of an improved maize

variety in the Machakos region of Kenya. The modeling approach described above is

implemented in the Tradeoff Analysis Model for Multi-Dimensional Impact Assessment

(TOA-MD5.0) model. The TOA-MD5.0 model, model documentation, and the dataset

used for this analysis, are available at http://tradeoffs.oregonstate.edu.

This case study is of interest because agricultural technology and its impacts on

the poverty and sustainability of agricultural systems in Kenya and elsewhere in Sub-

Saharan Africa are the subject of much concern, given adverse trends in these indicators

over the past several decades (Lesschen et al. 2007; Barrett 2008). Kenya has been the

subject of much recent research on these issues, particularly with respect to the

cultivation of maize (Marenya and Barrett 2009; Suri 2011). This case study also

illustrates how a parsimonious impact assessment can be implemented at low cost using

existing survey data, similar to data that have been collected in many research projects

around the world. These data were originally collected as part of a project on sustainable

nutrient management that has been documented elsewhere in detail (Gachimbi et

al.2005), that was not designed to address the impact of high-yielding maize varieties.

In this region the predominant system is small-scale, semi-subsistence crop and

livestock agriculture. Machakos is a diverse region, and in some areas where irrigation is

available, vegetable production is the dominant system and provides higher incomes.

Some farms also produce milk for sale and tend to have higher incomes than farms that

rely mostly on subsistence crops. Accordingly, in this case study the population was

stratified into three groups: farms growing only subsistence crops; farms producing a

21

substantial amount of dairy along with subsistence crops; and farms with irrigated

vegetables. About 30 percent of cropped area is planted to maize in the subsistence

group, compared to less than 20 percent for the other two, and agricultural incomes per

farm are substantially higher for the latter two groups, as shown below. Data from the

region show that poverty is high, with the headcount poverty rate around 75 percent

(based on a $1/person/day poverty line). The sustainability of the system is low, as

indicated by high rates of soil nutrient and organic matter losses.

In this case study, I assume that the improved maize variety results in a doubling

of average yields and lowers the spatial standard deviation in maize returns by 25

percent, holding the distribution of prices constant – assumptions consistent with the

estimates of the yield effects of hybrids presented by Suri (2011) for Kenya and studies in

other East African countries (e.g., Alene and Hassan 2006), and the data on which this

analysis is based. I also assume that in response to this change in yield, farmers double

the amount of land allocated to maize and reduce other crops proportionately. The

assumed yield increase is technically feasible, given that the genetic potential for maize

exceeds 10 tonnes/ha, and that Machkos maize yields are extremely low, averaging less

than 2 tonnes/ha in the 6 villages represented in the study and less than 1 tonne/ha in

some areas. Farmers’ use of organic and mineral fertilizer is also extremely low, resulting

in rates of soil nutrient loss that are estimated to average about 30 kg/ha/season. The

correlation ρv cannot be observed in this prospective analysis, but as explained above, we

can logically deduce that this correlation should be positive and relatively close to 1 in

this case. This high correlation is expected because maize yields of conventional and

high-yielding varieties are similarly impacted by environmental conditions such as

22

weather and soils, and because other components of the system are similar. A sensitivity

analysis investigates this assumption below.

Figure 3 presents the simulated adoption curves for the three farm types and all

farms, derived by varying the adoption threshold a (see equation 1). The predicted

adoption rates for the high yielding maize variety (the point where the curve crosses the

horizontal axis) are about 70, 63 and 57 percent for subsistence, dairy and vegetable

farms, implying a 66 percent rate in the overall population, consistent with Proposition 1.

These predicted adoption rates are similar to the observed rate of adoption of hybrid

maize varieties in Kenya (Suri 2011), thus demonstrating the ability of the model based

on heterogeneous expected returns to predict adoption rates consistent with actual

adoption rates. Figures 4 and 5 show the behavior of mean returns for non-adopter and

adopter groups for subsistence and vegetable farms (dairy farm results are similar to

vegetables and are not shown). The graphs show a pattern of mean returns for non-

adopters and adopters consistent with the theoretical predictions of Proposition 2. Mean

returns in the population attain a maximum, consistent with Proposition 3 and Corollary 1

(note that the maxima of the net returns for entire population do not occur at the

intersection of the adopter and non-adopter curves in the upper panel because the

population mean is the weighted sum of the sub-population means, see equation 5 and the

Appendix). The poverty rate predicted by the model for the entire population is close to

the observed rate of about 75 percent.

Let the mean nutrient loss indicator for the entire population be In(a) = r(1,a)

In(1,a) + r(2,a) In(2,a) where In(h,a) = µn(h,a) is the population mean soil nutrient loss for

system h. Define a threshold indicator In(a,τn) = r(1,a) In(1,a,τn) + r(2,a) In(2,a,τn), where

23

In(h,a,τn) = ∫ χ𝛕𝛕𝒏𝒏(n|h,a) dn, with τn defined as farms with soil nutrient losses less than 20

kg/ha/year. Table 1 presents the correlations estimated for the nutrient loss outcomes, and

the implied values of the correlations between nutrient loss and opportunity cost. Table 1

shows that the correlations with opportunity cost are all negative but differ substantially

in magnitude. The correlations for system 1 are less in absolute terms than system 2, due

to the fact that maize is a nutrient-depleting crop and more maize is grown in system 2.

The covariances for the irrigated vegetable system are an order of magnitude larger than

those for the subsistence and dairy systems, reflecting the higher yields obtained in the

irrigated areas. The between-system correlations are positive and between 0.8 and 0.9,

consistent with the argument made above that these correlations are generally large and

positive for similar systems.

Figure 6 shows the mean soil nutrient loss for non-adopters and adopters, and for

the entire population, for the irrigated vegetable farms. Figure 6 shows several important

facts: first, maize is a nutrient-depleting crop, particularly with the low levels of fertilizer

use typical of Machakos, so adoption of a high yielding maize variety results in higher

nutrient losses; second, the indicators for non-adopters and adopters vary systematically

with the adoption rate according to the correlation between opportunity cost and the

outcome variable (Proposition 2); and third, the indicator for the entire population is a

nearly straight line connecting In(1,-∞,τn) and In(2,+∞,2), because the covariances

between opportunity cost and the outcomes of each system are very similar (Corollaries 2

and 3; see Table 1).

Figures 7 and 8 present the threshold soil nutrient indicators for subsistence farms

and the farms producing irrigated vegetables. The indicators for the subsistence adopters

24

and non-adopters are approximately linear with similar, relatively low slopes, resulting in

a nearly linear aggregate relationship (Corollary 2). The adopter curve for the vegetable

farms in figure 8 is close to the horizontal axis, showing that most adopters have high

nutrient losses. In contrast, non-adopters have lower losses, so the slope of the non-

adopter curve is positive, and as a result the curve for the entire population of vegetable

farms is non-linear.

Table 2 reports results of a sensitivity analysis, using the adoption rate, poverty

rate and threshold nutrient loss indicators. As noted above, this analysis was based on

assumptions for the following parameters: between-system correlations; the change in

average maize yield associated with adoption of the high yielding variety; the change in

the maize returns standard deviation; and the average share of land allocated to maize in

the systems. The values specified above were treated as the middle of the plausible range.

Table 1 shows that the adoption rate and the poverty and nutrient loss indicators do not

deviate more than 14 percent when the parameters are varied individually from low to

high values. When the parameters are varied together, the adoption rate ranges from 56 to

81 percent; the largest change is a 27 percent lower value for the nutrient loss indicator.

This case study illustrates two important implications of the more general analysis

presented above. First, the predicted impacts for the non-adopter and adopter sub-

populations depend on the correlations between opportunity cost and the outcome

variables. Assuming zero correlations would be equivalent to assuming that the curves in

figures 4-8 for adopters and non-adopters are horizontal lines, implying large errors in

estimates of the indicators for both adopters and non-adopters. For example, in the case

of mean returns for vegetable farms, figure 5 shows that ignoring the correlations

25

between returns would result in an underestimate of returns at the predicted adoption rate

of almost 40 percent. For vegetable farms, figure 6 shows that the predicted mean

nutrient loss for system 2 at the predicted adoption rate is about 115 kg/ha/season;

ignoring the correlations between nutrient loss and opportunity cost, the predicted rate for

system 2 would be 75 kg/ha/season, an underestimate of about 35 percent.

Second, estimates of impact in the entire population of adopters and non-adopters

are likely to be less sensitive to correlations than the estimates of impact for each of the

sub-populations. This is the case for the examples presented in figures 6 and 7, because

the conditions of Corollary 3 are approximately satisfied. Additional research is needed

to investigate whether this finding is typical and thus can be generalized.

Conclusions and Extensions

The parsimonious, generic approach to multi-dimensional impact assessment presented in

this article responds to the growing demand for assessments that address economic,

environmental and social impacts of agricultural technologies, as well as assessments of

impact and adaptation to climate change and other environmental changes. To be useful,

impact assessment methods must be model-based and feasible within the time and

financial constraints of technology-related projects. The model presented here achieves

parsimony in data and in model design, by providing a generic model structure that can

be used for virtually any type of system, and that can be parameterized with low-order

moments of outcome distributions.

The analysis of this model shows that means, variances, and within- and between-

system correlations of adoption and outcome variables are needed for accurate estimation

of impacts in non-adopter and adopter sub-populations. Stakeholders often want to know

26

impacts on adopters and non-adopters, and the analysis presented here shows that to do

this it is important to have credible estimates of requisite correlations in addition to

means and variances. This finding could be interpreted as showing a need for more and

better data suitable to estimate correlations. Until now the importance of these

correlations has not been recognized, and there are few estimates of them in the literature,

so a call for better data is justified.

The parsimonious modeling approach advocated here suggests, however, that

costly new farm survey data is not the only answer to the challenge of multi-dimensional

impact assessment. The analysis presented here shows that it may be possible to carry out

sufficiently accurate analysis by putting reasonable bounds on parameters by using a

combination of a priori reasoning and available data. For example, the case study

presented in this article showed that the within-system system correlations for system 1

and system 2 had a logical relationship to each other, so that knowledge of system 1

could be used to place bounds on parameters of system 2. This study also showed that

between-system correlations are positive and relatively high when systems 1 and 2 are

similar, and that sensitivity analysis can be used to see if results are substantially different

over a plausible range of parameter values. Over time it should be possible to compile

estimates from case studies of various systems and regions, and use meta-analysis to

establish empirical parameter distributions to use in subsequent studies, thus reducing or

eliminating the need to collect new data for every impact assessment. In addition, in this

article I showed that, under certain conditions, correlations do not need to be known to

estimate aggregate impacts accurately. These considerations suggest that better use of

27

existing data, and investments in new data in those cases where available data are

inadequate, can lower the costs of carrying out multi-dimensional impact assessments.

The model presented in this article has been implemented as the Tradeoff

Analysis Model for Multi-Dimensional Impact Assessment (TOA-MD5.0). This model

is publicly available with documentation at http://tradeoffs.oregonstate.edu. The TOA-

MD model can be used to assess impacts in a variety of ways. First, it can be used to

carry out “tradeoff analysis” of technology adoption by simulating the relationships

between economic, environmental and social indicators as the adoption threshold and

other model parameters are varied. The model is also designed to simulate farmers’

participation in ecosystem service contracts and the resulting ecosystem service supply

curves, as in Antle and Valdivia (2006) and Antle et al. (2010). Another use is analysis of

climate change impacts and adaptation as presented in Claessens et al. (2010).

Several useful extensions of the model developed in this article are possible. One

would be to incorporate the temporal dimension of technology diffusion, and develop a

parsimonious empirical model with this feature. For example, the adoption threshold

could be specified as varying with time, thus inducing a time path for the adoption rate.

Another useful extension would be to explore the possibility of constructing a model

based on non-normal joint distributions of opportunity cost and outcome variables. Doing

so will most likely require complex computational methods (e.g., Nadarajah and Kotz

2006). Another worthwhile extension would be to link this type of impact assessment

model with a market equilibrium model, so that price effects of supply shifts associated

with technology adoption could be incorporated. The assumption used in the case study

presented here, that price distributions are the same for both systems, is a good starting

28

point for most impact assessments, and in many cases this assumption will be appropriate

because the adoption rate is low, or because the region being analyzed is small relative to

output and input markets. However, it is also possible to specify system 2 with the price

distributions that would be associated with a change in output caused by the adoption of

system 2. As always, researchers will have to judge the value of introducing this

additional complexity relative to the costs of time and other resources needed to

implement an impact assessment.

29

Footnotes

1. For example, the 2010 call for proposals on Gobal Food Security from the National

Institute for Food and Agriculture states, “For new CAP proposals, by the end of 5

years (or earlier), the project director and team are to report the estimated overall

economic impact of the CAP activities, as well as other significant, relevant outcomes

such as, but not limited to, behavioral, social, and environmental.” (National Institute

for Food and Agriculture, 2010, p. 10).

2. For example, in a recent ex post impact assessment of aquaculture in Malawi, Dey et

al. (2006, p. 32) reported, “While the analysis provided a good understanding of the

adoption process, no exact data on the scale of adoption were available. Hence, the

total number of [integrated agriculture-aquaculture] IAA practitioners and what

proportion of aquaculture production can be attributed to IAA had to be assumed.”

Similarly, in a recent study of the impacts of nutrient management research in the

Philippines, Walker et al. (2009) made assumptions about the rate and extent of

adoption in the study area and in other regions where adoption was expected to occur.

Walker et al. (2008, pp. 33-34), observe: “Compared to estimates on other variables

in an epIA (ex-post impact assessment) on agricultural research, those on adoption

are usually shrouded in uncertainty. Economic rate of return assessments are

predicated on annual estimates of adoption. It is only for very few technologies that

annual estimates can be furnished from primary or secondary data without having to

resort to projection or backward forecasting. Sensitivity analysis often shows that

estimates of the size of net benefits are more sensitive to adoption levels and rates

than to those of any other variable (Walker and Crissman, 1996).”

30

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36

Appendix

The purpose of this appendix is to derive the properties of the distributions of returns (v)

and environmental and social outcomes (z) associated with the sub-populations using

each system, indexed by k, as well as the distribution for the entire population composed

of non-adopters and adopters. Following the notation introduced in the article, if all farms

in a region use system h, then the joint and marginal distributions of v and z are defined

as φ(v,z | h) and φ(k | h), for k = v,z. These distributions are defined for given prices and

other exogenous variables.

The adoption variable ω and the outcomes k = v, z are jointly distributed in a

population. To construct these joint distributions, we can define the conditional outcome

distributions φ(k | ω,h) associated with each ω for system h, and then use Bayes’ rule to

obtain φ(ω,k | h) = φ(k | ω,h) ϕ(ω). Using equation (1), it follows that for the sub-

population of farms adopting system 2, where ω < a, we have:

(A1) ∬ φ𝑎𝑎−∞ (ω,k |h) dω dk = ∫ {∫ φ𝑎𝑎

−∞ (k |ω,h) dk} ϕ(ω)/r(h,a) dω

= ∫ ϕ𝑎𝑎−∞ (ω) dω = r(2,a).

Similarly, for the sub-population of farms using system 1, the integral of φ(ω,k | h) over ω

> a gives r(1,a). Therefore, the distribution for the sub-population using each system is

the joint outcome distribution between ω and k = v, z, truncated according to ω > a for

system 1 and ω < a for system 2:

(A2) φ(ω,k |h,a) ≡ φ(ω,k |h)/r(h,a) = φ(k |ω,h) ϕ(ω)/r(h,a).

Integrating (A2) over ω > a for system 1 and over ω < a for system 2, we have

(A3) χ(k |h,a) = ∫ φ(ω,k | h,a) dω = ∫ φ(k | ω,h) ϕ(ω)/r(h,a) dω.

37

The distribution of k = v,z in a population using both systems is a mixture of the

distributions defined in (A3), with mixing proportions r(h,a) (McLachlan and Peel 2000):

(A4) χ(k | a) = r(1,a) χ(k | 1,a) + r(2,a) χ(k | 2,a).

We can construct moments of these outcome distributions conditional on adoption

threshold a, by deriving the moments of the outcome variables conditional on ω, and then

taking the expectation with respect to ω. Using (A2) and (A3), the mean of k given a is:

(A5) µk(h,a) = ∫ 𝑘𝑘 χ(k |h,a) dk = ∬𝑘𝑘 φ(k |ω,h)ϕ(ω)/r(h,a) dk dω

= ∫{∫ 𝑘𝑘 φ(k |ω,h) dk}ϕ(ω)/r(h,a) dω = ∫µk(ω,h)ϕ( ω)/r(h,a) dω,

where µk(ω,h) is the mean of k given ω, for system h, and integration is over

ω > a for system 1 and ω < a for system 2. Following Johnson and Kotz (1970), under

normality the conditional mean of outcome k, given ω, is:

(A6) µk(ω,h) = µk(h) + (ω – µω) σk(h)θk(h)/σω,

where the parameters are defined in the article text above equation (6). For a standard

normal density φ*, the Inverse Mills’ ratio for the truncated distribution of ω associated

with each system is:

(A7) λ(a,h) = φ*[(a – µω)/σω]/r(h,a).

The means of the truncated distributions of ω for each system are

(A8) µω(h,a) = µω – (–1)h σω λ(a,h).

Taking the expectation of (A6) with respect to the truncated distribution of ω, and using

(A7) and (A8), it follows that the means of the truncated outcome distributions are:

(A9) µk(h,a) = µk(h) – (–1)h σk(h)θk(h)λ(a,h).

For a bivariate normal distribution, the variance of k conditional on ω has been shown to

be (Aitkin 1964):

38

(A10) σ𝑘𝑘2(h) {(1 – θk(h)2) + θk(h)2 (1 + (–1)h-1a λ(a,h) – λ(a,h)2)}.

Using (A9) and (A10), we can parameterize the distributions χ(k | h,a) and compute the

indicators defined in (4) and (5).

Proof of Proposition 2

Using (A3) and (4) we have

Ik(h,a,τ) = ∫ ι𝜏𝜏 (k) χ(k |h,a) dk = ∫ ι(𝑘𝑘)∫ φτ (k | ω,h) ϕ(ω)/r(h,a) dω dk

where the inner integral is taken over (a,+∞) for system 1 and (-∞,a) for system 2. Now

differentiate to obtain:

(A11) 𝜕𝜕I𝑘𝑘(ℎ ,𝑎𝑎 ,𝜏𝜏)𝜕𝜕𝑎𝑎

= (– 1h)λ(a,h)∫ ιτ (k)φ(k | a,h) dk

+ (– 1)h-1 λ(a,h)∫ ι(𝑘𝑘)τ ∫ φ(k | ω,h) ϕ(ω)/r(h,a) dω dk

= λ(a,h) (– 1)h-1{Ik(h,a,τ) – Mk(h,a,τ)},

To derive (A11), I use (A7), ∂r(h,a)/∂a = (– 1)hϕ(a), I interpret φ(k | a,h) as the

conditional distribution φ(k | ω,h) evaluated at ω = a, and

(A12) Mk(a,h,τ) ≡∫ ιτ (k) φ(k | a,h) dk

is the expectation of ι(k) for system h at ω = a. The next step is to observe that Mk(a,h,τ)

is the change in the value of Ik(h,a,τ) at a, holding r(h,a) constant . Thus, if ω and k are

positively correlated for system 1, then the marginal change in the indicator, Mk(a,1,τ), is

less than the average value of the indicator, Ik(1,a,τ), thus from (A11) it follows that

𝜕𝜕I𝑘𝑘(1,𝑎𝑎 ,τ)𝜕𝜕𝑎𝑎

> 0, with the inequality reversed if ω and k are negatively correlated. Similarly, if

ω and k are positively correlated for system 2, then we have Mk(a,2,τ) > Ik(2,a,τ),

implying 𝜕𝜕I𝑘𝑘(2,𝑎𝑎 ,τ)𝜕𝜕𝑎𝑎

> 0, with the inequality reversed if the correlation is negative. If ω and

39

k are uncorrelated then Mk(a,h,τ) = Ik(h,a,τ) and 𝜕𝜕I𝑘𝑘(ℎ ,𝑎𝑎 ,τ)𝜕𝜕𝑎𝑎

= 0. Equation (1) shows that

𝜕𝜕𝜕𝜕 (2,𝑎𝑎)𝜕𝜕𝑎𝑎

= ϕ(a) > 0, so the same results apply to dIk(h,a,τ)/dr(2,a) = 𝜕𝜕I𝑘𝑘(ℎ ,𝑎𝑎 ,τ)𝜕𝜕𝑎𝑎

/ 𝜕𝜕𝜕𝜕 (2,𝑎𝑎)𝜕𝜕𝑎𝑎

.

Finally, note that if τ is defined as an upper threshold, such as a poverty line, then the

sign of (A11) is reversed. Q.E.D.

Proof of Proposition 3

The goal is to show that Iv(a,τ) defined in equation (5) is maximized at the threshold

value a = 0, i.e., at the point where ω = v(1) – v(2) = 0. From (A3) and (A4), the

distribution of v given a is:

(A13) χ(v | a) = r(1,a) χ(v | 1,a) + r(2,a) χ(v | 2,a)

= ∫ φ∞𝑎𝑎 (v | ω,1) ϕ(ω) dω + ∫ φ𝑎𝑎

−∞ (v | ω,2) ϕ(ω) dω.

Iv(a,τ) can therefore be written as:

(A14) Iv(a,τ) = ∫ ιτ (v){∫ φ∞𝑎𝑎 (v | ω,1) ϕ(ω) dω + ∫ φ𝑎𝑎

−∞ (v | ω,2) ϕ(ω) dω} dv.

The first-order condition for a maximum of Iv(a,τ) is thus:

(A15) 𝜕𝜕I𝑣𝑣(𝑎𝑎 ,𝜏𝜏)𝜕𝜕𝑎𝑎

= ∫ ιτ (v){–φ(v | a,1) ϕ(a) + φ(v | a,2) ϕ(a)} dv

= ϕ(a){Mv(a,2,τ) – Mv(a,1,τ)} = 0

where Mv(a,h,τ) is defined in (A12). Observe that since ω = v(1) – v(2), at ω = 0 we have

v(1) = v(2), and therefore φ(v | 0,1)= φ(v |0,2), thus demonstrating that (A15) is satisfied

at ω = a = 0. Thus if the indicator is increasing (decreasing) for a < 0, this must be a

maximum (minimum). Differentiating (A15),

(A16) 𝜕𝜕2I𝑣𝑣(𝑎𝑎 ,𝜏𝜏)𝜕𝜕2𝑎𝑎

= ϕ′(a) {Mv(a,2,τ) – Mv(a,1,τ)} + ϕ(a) {𝜕𝜕M𝑣𝑣(𝑎𝑎 ,2,𝜏𝜏)𝜕𝜕𝑎𝑎

– 𝜕𝜕M𝑣𝑣(𝑎𝑎 ,1,𝜏𝜏)𝜕𝜕𝑎𝑎

}.

40

From (A15) we know that the first term on the right-hand side of (A16) is zero at a = 0,

so the sign of (A16) is determined by the sign of the second term, which is negative

(positive) as long as 𝜕𝜕M𝑣𝑣(𝑎𝑎 ,2,𝜏𝜏)𝜕𝜕𝑎𝑎

< (>) 𝜕𝜕M𝑣𝑣(𝑎𝑎 ,1,𝜏𝜏)𝜕𝜕𝑎𝑎

. Q.E.D.

Proof of Corollary 1

Substituting equation (11) into (A6) shows that under normality,

(A17) Mv(a,h,τ) = µk(h) + (a – µω) σk(h) {σv(1)ρvh-1 – σv(2)ρv

2-h}/σω2 .

Using (A17), it follows that Mv(a,2,τ) – Mv(a,1,τ) = – a, so (A15) is satisfied at a = 0 and

that the second order condition (A16) is satisfied for a maximum. Since expected returns

are therefore maximized at a = 0, it follows that aggregate returns are maximized. Q.E.D.

Proof of Corollary 2

From equation (1) we have ∂r(2,a)/∂a = ϕ(a). Combined with equation (A15) we have

(A18) dIk(a,τ)/dr(2,a) = 𝜕𝜕I𝑘𝑘(𝑎𝑎 ,𝜏𝜏)𝜕𝜕𝑎𝑎

/ 𝜕𝜕𝜕𝜕 (2,𝑎𝑎)𝜕𝜕𝑎𝑎

= Mk(a,2,τ) – Mk(a,1,τ).

Thus dIk/dr(2,a) is a constant if Mk(a,2,τ) – Mk(a,1,τ) is a constant. A sufficient condition

for this to be true is that 𝜕𝜕M𝑘𝑘(𝑎𝑎 ,2,𝜏𝜏)𝜕𝜕𝑎𝑎

= 𝜕𝜕M𝑘𝑘(𝑎𝑎 ,1,𝜏𝜏)𝜕𝜕𝑎𝑎

. Q.E.D.

Proof of Corollary 3

For mean indicators under normality, from (A6) we have 𝜕𝜕M𝑘𝑘 (𝑎𝑎 ,ℎ ,𝜏𝜏)𝜕𝜕𝑎𝑎

= σk(h)θk(h)/σω, and

thus if 𝜕𝜕M𝑘𝑘(𝑎𝑎 ,2,𝜏𝜏)𝜕𝜕𝑎𝑎

= 𝜕𝜕M𝑘𝑘(𝑎𝑎 ,1,𝜏𝜏)𝜕𝜕𝑎𝑎

as required by Corollary 2, it follows that the covariances of

k and ω are equal for both systems, i.e., σωσk(2)θh(h) = σωσk(1)θh(h). Q.E.D.

Proof of Corollary 4

Using (A12), differentiate Mk(a,h,τ) with respect to a, and then use (A18). Q.E.D.

41

Table 1. Estimated Correlations for Nutrient Loss Outcomes, Machakos, Kenya

Note: Cov(h) is the covariance between nutrient loss and opportunity cost for system h,

scaled by 106.

κn (1) κn (2) ρn θn (1) θn (2) Cov(1) Cov(2)

Subsistence 0.56 0.70 0.89 -0.02 -0.42 -0.15 -0.46

Dairy & Crops 0.37 0.45 0.89 -0.05 -0.20 -0.13 -0.77

Irrigated Veges 0.00 0.54 0.81 -0.67 -0.82 -7.06 -7.17

42

Table 2. Sensitivity Analysis of Selected Model Parameters for Subsistence Farms in

Machakos, Kenya, Adopters of Hybrid Maize

Note: Low = simulation with low parameter value, high = simulation with high parameter

value. Numbers in bold are low or high values divided by the mid value. Poverty rate

defined as the percent of population below $1 income per person per day. Nutrient loss is

the proportion of farms with losses less than 20 kg N/ha/season.

Mid-valueLow High Low High Low High

Between-system correlation 63.2 74.5 67.8 71.9 13.3 10.3 0.93 1.09 0.97 1.02 1.13 0.87Mean maize yield 59.6 74.1 78.6 62.5 12.5 11.5

0.88 1.09 1.12 0.89 1.06 0.97Maize returns std deviation 70.2 65.8 74.1 65.8 13.5 10.1 1.03 0.97 1.06 0.94 1.14 0.86Maize land share in system 64.6 71.2 73.6 65.5 12.0 11.4 0.95 1.05 1.05 0.93 1.02 0.97Combined effects 53.8 86.9 75.2 60.9 13.0 10.5

0.79 1.28 1.07 0.87 1.10 0.89

Parameters: Low Mid HighBetween-system correlation 0.6 0.8 0.9Mean maize yield +50% +100% +150%Maize returns std dev +0% -25% -50%Maize land share in system 0.4 0.6 0.8

68.1 70.2 11.8Adoption rate Poverty rate Nutrient loss

43

Figure 1. Example of the Effects of Partial Adoption of System 2 on the Outcome

Distributions for Adopters, Non-Adopters and the Entire Population.

Entire Population with adoption: 60% > τ

r(1,a) non-adopters

System 1: 30% > τ

System 1 before adoption: 40% > threshold τ

System 2: 85% > τ

τOutcome k

φ(k|1)

r(2,a) adopters

χ(k|1,a) χ(k|2,a)

χ(k|a)

44

Figure 2. Contours of Equal Density for the Joint Distribution of Opportunity Cost ω and

Outcome Variable k for systems 1 and 2.

Note: µk(h,0) is the mean indicator for system h with adoption threshold a = 0. For an

indicator defined as the percent of outcomes exceeding threshold τ, areas c + d + e + f

represent the threshold indicator for system 1 before adoption of system 2 (ignoring areas

outside the contour as negligible). After adoption of system 2, areas e + f represent the

threshold indicator value for system 1 and areas b + d represent the value for system 2.

ω

k

0

•µk(2,0)

τµk(1,0)

•

System 1

System 2

b

c d e f

45

Figure 3. Adoption Curves for High Yielding Maize Variety, Machakos, Kenya

-200000

-150000

-100000

-50000

0

50000

100000

0 10 20 30 40 50 60 70 80 90 100

Opp

ortu

nity

Cos

t (Ks

h/fa

rm)

Adoption Rate (%)

Subsistence Crops Dairy and Crops Irrigated Vegetables and Crops All Farms

46

Figure 4. Mean Farm Income of Machakos Subsistence Farms with Adoption of a High-Yielding

Maize Variety.

0

20000

40000

60000

80000

100000

120000

0 10 20 30 40 50 60 70 80 90 100

Mea

n Ag

ricul

tura

l Inc

ome

per F

arm

(Ksh

/sea

son)

Adoption Rate (%)

Subsistence Crops - Non-adopters Subsistence Crops - Adopters Subsistence Crops - All Farms

47

Figure 5. Mean Farm Income of Machakos Irrigated Vegetable Farms with Adoption of a High-

Yielding Maize Variety.

0

50000

100000

150000

200000

250000

300000

0 10 20 30 40 50 60 70 80 90 100

Mea

n Ag

ricul

tura

l Inc

ome

per F

arm

(Ksh

/sea

son)

Adoption Rate (%)

Vegetables & Crops - Non-adopters Vegetables & Crops - Adopters Vegetables & Crops - All Farms

48

Figure 6. Mean Nutrient Loss for Machakos Irrigated Vegetable Farms with Adoption of a High-

Yielding Maize Variety.

-50

0

50

100

150

200

250

0 10 20 30 40 50 60 70 80 90 100

Mea

n S

oil N

utrie

nt L

oss

(kg/

ha/s

easo

n)

Adoption Rate (%)

Non-Adopters AdoptersAll Farms Non-Adopters - Zero CorrelationsAdopters - Zero Correlations

49

Figure 7. Percent of Farms with Nutrient Loss < 20 kg/ha/season, for Machakos Subsistence

Farms with Adoption of a High-Yielding Maize Variety.

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70 80 90 100

Farm

s with

Soil

Nutr

ient

Los

s < 2

0 kg

/ha/

seas

on (%

)

Adoption Rate (%)Subsistence Crops - Non-adopters Subsistence Crops - Adopters Subsistence - All Farms

50

Figure 8. Percent of Farms with Nutrient Loss < 20 kg/ha/season, for Machakos Irrigated

Vegetable Farms with Adoption of a High-Yielding Maize Variety.

0

10

20

30

40

50

60

70

80

90

100

0 10 20 30 40 50 60 70 80 90 100

Farm

s with

Soil

Nutr

ient

Los

s < 2

0 kg

/ha/

seas

on (%

)

Adoption Rate (%)

Vegetables & Crops - Non-adopters Vegetables & Crops - Adopters Vegetables & Crops - All Farms