262 J.R. Anderson Agriculture, Ecosystems and Environment 82 2000 261–271
issues to ensure effective and relevant linkages across social and environmental as well as other more tradi-
tional spatial and investment considerations, in what the Bank calls Environmentally and Socially Sustain-
able Development.
Clearly it is not just the changing perception of the importance of supra-national issues that has facilitated
such broader purviews. The technology now avail- able to analysts has greatly expanded the opportuni-
ties, and these are being increasingly seized. GIS and GPS, electronic computers, contemporary software,
and modeling capability generally, have all surely led to significant expansion in the capacity for well-tuned
even inexperienced analysts to do so much more than their parents could ever have hoped to do at higher
levels of both resolution and aggregation.
And yet life is still not necessarily all so easy. The misleading nature of assessment of sustainability ques-
tions at crop-plot or field level, while ignoring other linked elements of the landscape, such as forests and
less-agriculturally managed areas Leach and Mearns, 1996, has given greater recognition of the need for
broad spatial scales of analysis, such as at the ecosys- tem Gregory and Ingram, 2000 or catchment Tinker
and Anderson, 1996 levels. This is overtly the case for global-level issues such as concerning the atmo-
sphere, but analogously pertains too to the geosphere and biosphere.
The problematic of this paper can now be stated. Many research workers have their hands-on inves-
tigative activities primarily at the level of field plots or process models, and yet are interested in saying
something sensible perhaps to policy makers about the behavior of some defined “universe”, at a wider
or more aggregate level of a system. To do this they must use a process such as addition, multiplication,
aggregation, magnification, transformation or general- ization. Within this unsettled set of procedures, there
are possibilities of combinations of method, and op- portunities for error, such as fallacy of composition;
in short, the challenge of the scaling-up problem.
For agricultural resource analysts, scaling up seems to be mainly concerned with the representativeness of
plot-level observations for making statements about the status of a resource such as soil in a patch-mozaic
agriculture Scoones and Toulmin, 1999, p. 29, 31, 41. Agricultural ecologists take this idea to a higher
level of generality Fresco and Kroonenberg, 1992. Engineers call scaling work dimensional analysis,
and there are well agreed procedures for particular classes of scale models. For yet others, the scaling-up
problem is more one of the difficulty of getting widespread adoption of worthy technical solutions
that seem to work well in local “islands of success” in developing agriculture Pretty, 1995, 1997. More
specifically, in the context of World Bank operations, it usually means handling the challenge of moving
from a pilot operation to a “full-scale” project. In- deed, the World Bank President has recently opined
that the most pressing problem facing Bank staff and their clients is that of scaling up from project-level
activities that work well enough at that typically more bounded level, to wider endeavors that really make a
difference in poverty alleviation at the national level and beyond; or else “the race” is being lost.
For the level of generality sought in this overview, it may be that there are no general fixes widely appli-
cable to diverse phenomena. To consider briefly one of the more general treatments observed, White et al.
1998, Table 19.3 offer tantalizing advice — tanta- lizing because there is no further discussion of these
“problem-solving” suggestions in the paper — about “scaling up” under three heads:
synthesis across sites — GIS and modeling these two data sourcesmethods for all the three, site
similarity studies; extrapolation of practices — extrapolation domain
definition; and links among system levels — watershed studies,
decision support systems. The sections pertaining to response and risk that
follow set out considerations that point to some pos- sibly useful procedures. Needless to say, adequate
scaling-of-model insight is a necessary but not a suf- ficient condition for good policy making at global
ecosystem levels, but the latter large topic is beyond the scope of this paper, covered herein by Norse and
Tschirley 2000.
3. Response and aggregation
3.1. Conceptual Many cases of inference to high levels of aggre-
gation involve scaling up from a response process
J.R. Anderson Agriculture, Ecosystems and Environment 82 2000 261–271 263
function f linking a set of micro-level variables y = f x, z, u,
1 where y is output, x a controlled input, z an uncon-
trolled factor, and u is a random term, to a universe of macro-level process function F and corresponding
variables linked in
Y = F X, Z, U , 2
and where all variables can be thought of as vec- tors. The main scaling concern in this paper is spatial.
Needless to say, it is not just the response relations themselves, but the findings that arise from their in-
terpretation that hold interest for the analyst. One can, with difficulty, imagine a world of perfect information,
where the random or disturbance terms could be omit- ted, as indeed they are largely in this section. In gen-
eral, however, they will have to be considered, if for no other reason than the Uncertainty Principle of Mod-
eling: “Refinement in modeling eventuates a require- ment for stochasticity” Mihram, 1972, p. 15. This
Principle and its application are pursued more vigor- ously in the second main section, which is addressed
explicitly to risk. For the present purpose, it is taken as given that analysts are primarily interested in what
happens on average in a response process of this type, so that analytic endeavor in this section is focused on
reliable estimation of the means or expected values of the respective variables. To the extent that this may re-
quire probability specification, methods such as those alluded to in the later section on risk will be needed.
It must be recognized, however, that many response processes do involve the interventionparticipation of
decision makers who may not be indifferent to risk which happily generates employment for applied so-
cial scientists, such as agricultural economists. Most people most of the time, in fact, are technically averse
to risk, and thus as individuals or in groups act other than simply to maximize the mean value of an objec-
tive function. So, even if the “scaling-up analyst” is prepared to ignore risk as is assumed in this section,
the fact that some or all of the agents whose behavior is being represented implicitly or explicitly in the pro-
cess of aggregating micro-relations to macro means that risk and risk aversion are inherent in the process,
and thus are only ignored at the peril of the analyst. 3.2. Practical
Many different approaches are available to attempt to aggregate from the means of Eq. 1 to those for
2. Schneider 1994, Chapter 14 discusses six scal- ing strategies used in ecology: i use a multiplication
factor simple adding being a special case; ii use limited-scope bounds; iii use a variable with large
scope to calculate a variable of more limited scope; iv use statistical scaling-up; v use H.A. Simon’s hier-
archy theory based on human organizations; vi use the principle of similitude. After reviewing the meri-
ts and problems with each, he sides with two other groups of authors led by E.B. Rastetter and J.A. Weins,
respectively, in recommending for practical purposes a combination of strategies built around v and i,
with supplementary use of dimensional reasoning of vi and good judgment, resorting to statistics only for
verification. The temptation is resisted here to fiddle with fractal geometry and illustrate the insights of al-
lometric relationships, such as y = ax
b
, to focus on metabolism e.g., West et al., 1999, and to explore
scaling issues as to why bats can and humans have trouble flying Costanza, 1991, p. 54.
The approach taken here and thus tentatively that recommended is surely more simplistic, being largely
a mix of strategies i and iv. If the functions are all linear or polynomial, and simply additive in the dis-
turbances, there is relatively little challenge involved in adding up the individual functions. But even in
this case, the matter may be non-trivial. Perhaps the best-known setting of the aggregation problem omit-
ting the disturbances is that concerning the use of the “representative farm” in prediction of aggregate agri-
cultural supply response. There is a large literature on this problem, e.g., Day, 1963; Buckwell and Hazell,
1972; Day and Sparling, 1977; Hanf and Mueller, 1979, but analytic attention in this field has shifted
to more encompassing usually price-endogenous non-linear models of agricultural sectors Harvey,
1990; Anderson et al., 1997. The risk aversion of producers in these early programming models was
captured through constraints to farm adjustment over time, rationalized as a presumed reluctance on the
part of producers to change too rapidly or too dras- tically from their previous farm plans, on grounds
of the costs of change, and of the changed risks faced. The essence of the scaling issue dealt with in
264 J.R. Anderson Agriculture, Ecosystems and Environment 82 2000 261–271
such models is that, while aggregation of technical relationships may be relatively straightforward, such
simplicity does not hold for aggregating behavioral relationships.
For more general than the linearpolynomial micro-production response relationships, the mathema-
tics of adding up can imply rather restrictive oppor- tunity for direct formal aggregation, even for some
of the simplest cases and where the disturbances are dropped. The subject is treated in depth by Chambers
1988, pp. 182–202, who in discussing aggrega- tion over optimizing firms, details the few classes
of function that can be aggregated, along with their confining mathematical properties technically, ho-
mothetic functions under conditions of separable pro- duction. The situation is considerably more complex
when the uncertainty terms are included, particularly when these are not merely additive homoscedastic
constant-variance error terms, but depend on the levels of the real variables x and z. Although seldom
recognized in empirical work, such heteroscedastic processes are likely to be the norm in applied pro-
duction processes e.g., Just and Pope, 1978; Griffiths and Anderson, 1982; Anderson and Hazell, 1994
because typically both controlled and uncontrolled inputsfactors influence risk experience. Thus unless a
high price of narrow mathematical and zero stochastic specification is paid, little can be done to aggregate
algebraically even simple response functions under general conditions. It does seem, however, that ana-
lytic aggregation at this level of generality has seldom, if ever, been done. Among the difficulties that may
have discouraged such endeavor would be the neces- sity for accounting for the risk aversion of producers
Dillon and Anderson, 1990, p. 154, if indeed the aggregate models were to represent a direct summa-
tion of the results of their optimizing behavior. The difficulty hinges on the greater challenge in aggregat-
ing behavioral response relationships relative to that faced in aggregating technological relationships. This
is doubtless why agricultural economists in dealing with aggregate risk-responsive behavior have usually
elected to work directly with aggregate data in, say, supply response analysis Dillon and Anderson, 1990,
pp. 181–184.
There is one special case of plot-to-regional scal- ing that seems deserving of a revisit from the present
vantage point, which was brought to mind in contem- plating the free-air carbon-dioxide enrichment FACE
investigations e.g., Jamieson et al., 2000. The late Bruce Davidson was struggling to use the available but
sparse experimental data to draw implications about commercial farm yields in a then-hypothetical Ord
River irrigation scheme in northern Western Australia. He observed a systematic non-linear association be-
tween matched experimental plot and nearby farm crop yields that was similar across a range of crops and
ecologies Davidson, 1962. With colleagues, these results were successfully generalized in a production
economics setting, which more recently would have been categorized as a meta-production relationship,
that emphasized and accounted for the differing in- tensities of inputs, such as labor and capital, beyond
those formally under test in the experiments David- son and Martin, 1956; Davidson et al., 1967. This
bit of history is mentioned because the same issue haunts any generalization of fx to FX, in the styles
of Eqs. 1 and 2, if the corresponding bundles of the z, Z and u, U vectors are not analogously mea-
sured and accounted for in the scaling up. A seeming gap in the scaling-up literature, however, is a careful
empirical quantification of the relationship between experimentally measured changes in as opposed to
merely levels of partial productivity measures associ- ated with, say, varietal change see also Alston et al.,
1995, pp. 338–340.
One of the obvious general approaches to aggrega- tion is to resort to sampling theory, although it seems
remarkably underused in this field of potential appli- cation. The essence of such an approach is the idea
that the plots that form the basis of the sample from which inferences are drawn about the relevant popu-
lation should be “appropriately” e.g., Scheaffer et al., 1996, p. 408 random and thus representative elements
of the defined populationuniverse, so that probabil- ity statements may be legitimately made about the
inferred estimates of the population parameters, espe- cially in mainstream practice, means and proportions.
Perhaps it is the demanding nature of the mechanisms that are used in sample survey design to ensure the
randomness of the observations that makes use of such an approach seemingly unconventional in generaliza-
tions from plots and models to a population level. It is, however, the logical reference case against which
any alternative method should be judged, although any such comparison appears to be rare in practice.
J.R. Anderson Agriculture, Ecosystems and Environment 82 2000 261–271 265
Where spatial probably GIS-based knowledge of some of the Z variables permits, a scheme of stratified
random sampling could usefully be used, in order to ensure deliberate representativeness as well as greater
precision in the sample design. Indeed, working with such ideas about the structure of the defined universe
rapidly takes one into a second statistical approach to seeking efficient extrapolation, namely the use of
principles of experimental design e.g., Mendenhall, 1968.
Several principles are relevant, without compromis- ing the key roles of randomness and representative-
ness. One is blocking, wherein systematic allocation of treatments i.e., specified combinations of X vari-
ables among blocks, perhaps defined in terms of some of the Z variables, enables the treatment effects to be
observed more precisely for a given size of sample. Another is partial factorial combination, perhaps of
both X and Z variables, to accomplish an efficient and parsimonious coverage of the space of interest, by us-
ing designs such as the central composite. These lead naturally to evolutionary designs, wherein with the
help of the digital computing revolution, automatic sequencing of a succession of experimental designs
takes the analyst to maximal or other optimal settings of the X response variables.
Statistics thus has a clear place in the aggrega- tion of plot-level data and relationships to a defined
universe, and thus underlines the propriety of includ- ing a stochastic disturbance term in the formulation.
But what else works? It has been noted that algebra per se is of limited utility. Programming models are
easier and have a reasonable record of achievement, provided that the disturbanceuncertainty terms are
not of interest. But on a priori grounds, the best gen- eral approach in terms of workability must reside
in ad hoc digital simulation methods, resorting as necessary to Monte Carlo sampling methods, as are
taken up in Section 4. While workable in general, such methods necessarily involve explicit and prob-
ably adaptive modeling of the aggregation process; what some authors describe as “teaching” or “tuning”
the aggregate model. Evidently there is consider- able scope for analyst ingenuity in using weighting
data at different levels of resolution, as revealed by some of the discussions in this CGTE Theme
such as Jamieson et al., 2000, and a critical need for ground-truthingcalibration in some cases, such
as in soil carbon and soil organic matter modeling Schlesinger, 1999.
One final subtlety might be noted. This is an aspect of the “adding-up” problem of aggregate relationships,
whereby the underlying resource situation is carefully captured in the modeling procedure, well illustrated in
recent work by Cynthia Rosenzweig and colleagues at NASAGoddard Institute on the treatment of water
dynamics in crop production. In thinking about such issues, one is reminded of the saga of the Limits to
GrowthWorld Dynamics modeling of the early 1970s, and the mis-specification of many key technology
and resource-supply dynamics e.g., Rothkopf, 1976 and embodied inconsistencies and fallacies e.g.,
Zeckhauser, 1973.
On resource considerations in adding up matters, at least in the context of examining agricultural produc-
tivity changes at the aggregate level, it may be salutary to return as did Alston and Pardey, 1996, p. 129 to
some early work of Schultz 1956 and his concept of an ideal ratio of “all outputs” to “all inputs” being ap-
proximately unity. It seems likely that many studies of agricultural productivity have overstated the growth
of multi-factor productivity and thus also the implied returns to investment in agricultural research because
of a tendency for technologies developed to involve a faster rate of exploitation of unmeasured components
of the natural resource stock, and thus an understate- ment of the flow of inputs Alston et al., 1995. As
has usually been the case, Schultz is doubtless right, and many incautious practitioners have probably been
at fault in their findings about the aggregate impact of apparently relevant plot-level response data. The
caution here is thus for analysts to do a more compre- hensive job in accounting for all the real inputs into a
response process, before claiming too much achieve- ment in productivity growth at the aggregate level.
3.3. Policy Policy relevance is very much in the eye of the
beholder, and the process of policy development as for sausage making sometimes best not beheld The
observations above, if valid, imply that, even putting aside matters of risk, there is still a way to go in the
development of inferential procedures that make gen- eralizations at the level of a defined universe a more
satisfying and comfortable procedure. Whether the
266 J.R. Anderson Agriculture, Ecosystems and Environment 82 2000 261–271
state of the art has advanced to the point where any sort of standard can be imposed is moot, although
the Schulze 2000 exposition may lead to a more common language.
While the progress that has been made is certainly something to be excited about, this observer remains
pessimistic about the robustness of some of the interim findings, and the operative call does indeed seem to be
one for further self-critical research. Some of the rese- arch needed concerns best use of the new tools them-
selves, especially the information-intensive GIS meth- ods e.g., Wood et al., 1999. For others the needed
research is at the biophysical process level itself, as indicated by the Schlesinger 2000 discussion of car-
bon sequestration and, for a rather different example, by the pesticide resistance dynamics relationships in
different forms of IPM e.g., Archibald, 1988; Schill- horn van Veen et al., 1997. Lest it seem that there
is some prejudice here against the natural sciences, it should be noted that there are ample controversies to
be resolved among the social sciences. One bridging issue relates to the aggregation concept of “gross natu-
ral product”, especially as a relevant indicator when it comes to environmental policy e.g., de Groot, 1992,
p. 254. To take another rather different case relevant to a discussion of agricultural aggregation, consider
Cochrane’s Treadmill Hypothesis. Cochrane 1958 viewed farmers as the victims of technological change,
because only early adopters of new and more produc- tive innovations would benefit and then only briefly,
as the increased output induced by the improved productivity depressed prices assuming inelastic
aggregate demand, and while consumers benefited from lowered commodity prices, farmers had to keep
running the treadmill to try to survive. Fortunately, the grain of truth in this hypothesis does not apply in
many if not most instances, as persuasively argued by Alston and Pardey 1996, p. 180. But it is instructive
to remind evaluators of applied agricultural research, as they seek to aggregate up from perhaps plot-level
estimates of improved agricultural productivity occa- sioned by some research activity to estimates of the
returns to research investment, that empirical issues of market structure are key to determining the nature
and distribution of the benefits of research — a topic dealt with in instructive detail by Alston et al. 1995.
It would be inappropriate to conclude this section on other than a positive note, however. The encour-
aging thing for the evolution of better policy analy- sis concerning world agriculture, such as alluded to
by Alexandratos 1995 and Anderson 1995, is that there is now real dialogue among policy makers work-
ing at the regional and global levels, and the existence of cogent scaled-up models, notwithstanding the vary-
ing degrees of refinement, has surely informed and facilitated such dialogue. But there is room for im-
provement in practice.
4. Risk and aggregation
