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Carryingcapacityinagriculture:Globalandregionalissues

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Ecological Economics 29 (1999) 443–461

ANALYSIS

Carrying capacity in agriculture: global and regional issues

Jonathan M. Harris *, Scott KennedyGlobal De6elopment and En6ironment Institute, Cabot Center, Tufts Uni6ersity, Medford, MA 02155, USA

Received 12 May 1997; received in revised form 10 August 1998; accepted 18 August 1998

Abstract

Much of the debate over world agricultural futures centers on the issue of yield growth. ‘Optimists’ in the field basetheir analyses on a technology- and input-driven rate of increase in yields which, they argue, has outpaced populationgrowth and will continue to do so, leaving a margin for steady increase in per capita incomes. Those who aresometimes classified as ‘pessimists’ (we prefer the term ‘ecological realists’) see practical limits to global and regionalcarrying capacity in agriculture, and maintain that the world is close to, or may even have passed, these limits.Projections for world agriculture in the first half of the twenty-first century vary widely, largely depending onassumptions on yield growth. We investigate the pattern of yield growth for major cereal crops, and present evidencethat the growth pattern is logistic, not exponential. This pattern is consistent with ecological limits on soil fertility,water availability, and nutrient uptake. Projections for supply and demand in the twenty-first century based on alogistic rather than an exponential model of yield growth imply that the world is indeed close to carrying capacity inagriculture, and that specific resource and ecological constraints are of particular importance at the regional level. Asupply-side strategy of increased production has already led to serious problems of soil degradation and wateroverdraft, as well as other ecosystem stresses. This implies that demand-side issues of population policy and efficiencyin consumption are crucial to the development of a sustainable agricultural system. © 1999 Elsevier Science B.V. Allrights reserved.

Keywords: Carrying capacity; Agricultural yields; Ecological limits; Population

1. Introduction

Is the world approaching carrying capacity inagriculture? A number of studies, including sev-eral by Worldwatch Institute (Brown, 1994, 1995,

1997) and our own article in Ecological Econom-ics (Harris, 1996), have suggested that the answeris yes. On the other hand, studies published by theWorld Bank and the Council for AgriculturalScience and Technology (CAST) are generallyoptimistic that meeting future food needs will bepossible and even ‘increasingly easy’ (Waggoner,1994; Mitchell and Ingco, 1995). Some analysts,

* Corresponding author. Tel.: +1 617 6285000, ext. 5470;fax: +1 617 6273712; e-mail: [email protected]

0921-8009/99/$ - see front matter © 1999 Elsevier Science B.V. All rights reserved.

PII: S0921-8009(98)00089-5

J.M. Harris, S. Kennedy / Ecological Economics 29 (1999) 443–461444

such as David Norse (1994) take a middle posi-tion, arguing that predictions of unprecedentedfood security crises are excessively simplistic, butthat technological optimism understates the im-portance of ecological stresses. Norse does, how-ever, imply some carrying capacity limit, which heestimates to be in the range of 12 billion people.Studies by the UN Food and Agriculture Organi-zation (FAO) and the International Food PolicyResearch Institute (IFPRI) see no global foodshortfalls at least through 2010, but note increas-ing problems of food security at the regional andcountry level (Alexandratos, 1995a,b; Agcaoiliand Rosegrant, 1995).

The wide divergence in projections of agricul-tural futures can be traced to the different method-ological perspectives of ecological and neoclassicaleconomics. Neoclassical models are oriented to-ward incremental growth without inherent limits;ecological models start from the premise that thereare inherent limits to the capture and use of solarenergy and planetary resources. The purpose ofthis paper is to explore some implications of thesemethodological differences, and to present someinitial empirical evidence on regional trends inagricultural productivity in the major cereal cropsof maize, wheat, and rice. We will argue that thesetrends are more consistent with the existence ofecological limits than with models based on tech-nology- and input-driven growth. In our conclu-sion, we will discuss some of the policyimplications of the introduction of ecological lim-its into models of agricultural growth.

2. Logistic versus exponential growth patterns

Econometric models, such as those employedby contributors to the IFPRI study, generallybase future agricultural production estimates on aprojected rate of yield growth. This yield growthis presumed to be a result of continuing techno-logical improvement and investment in agricul-ture. Historical growth rates are used as a baselinefor estimating future growth rates, although theestimated future growth rates may be lower thanhistorically observed rates, depending on themodel. The result is that these models generally

display exponential growth in yields over time.This is the crucial factor from which their mostlyoptimistic conclusions about future supply/de-mand balance are derived. Since most populationprojections show population following a logisticgrowth path towards eventual stabilization, evena modest sustained exponential rate of growth inyields will provide a comfortable, and increasing,margin over population growth, thus accommo-dating increased per capita consumption. In thesemodels, no ceiling or carrying capacity limit ap-pears. Their conclusions, accordingly, flow di-rectly from their methodological assumption ofexponential growth in yields.

An approach more oriented to the concept ofcarrying capacity limits would suggest, instead, alogistic path for crop yields, with some upperlimit imposed. In its early stages, a logistic growthpath closely resembles an exponential path. But asthe upper limit starts to exert more influence, thegrowth rate slows, passes through an inflectionpoint, and ultimately approaches zero as the car-rying capacity is approached. This suggests thateconometric modelers may be misled by an appar-ently exponential pattern of yield increase, failingto discern an incipient logistic pattern, an errorwhich would have increasingly severe conse-quences as the time period under considerationwas increased.

Of course, the seriousness of such an errorwould depend on the upper limit in question. PaulWaggoner, one of the most optimistic forecasters,does use a logistic projection for maize yields(Waggoner, 1994). For his upper limit on maizeyields, he uses a yield of 21 metric tons perhectare (t/ha), which is close to the theoreticalphotosynthetic limit on yields, and is about threetimes the average maize yield in the United Statestoday. His justification for this high limit is thatagricultural contest winners have actuallyachieved this yield level, proving it to be techni-cally possible. If we make a general assumptionthat ultimate limits on cereal yields are three timespresent US levels—given that the developingworld currently has average yields less than one-half of US levels—the resulting factor of aboutsix gives plenty of room to accommodate thedemands of a population of 8–11 billion, which is

J.M. Harris, S. Kennedy / Ecological Economics 29 (1999) 443–461 445

Fig. 1. Total cereal yields for Argentina, 1961–1995.

roughly the range envisioned in UN populationprojections for 2050 (United Nations, 1997).

Probably most people will react—as the au-thors of this paper do—with extreme skepticismto Waggoner’s assumption that such high yieldlevels could ever be achieved in practice, as aver-age yields over large regions. The theoretical ge-netic potentials of plant physiology are commonlyconstrained by unfavorable physicochemical envi-ronments (Boyer, 1982). Record yields such asthose cited by Waggoner have generally takenplace on soils with no significant productivitylimitations; but Boyer finds that only 12% of USsoils are in this category. Global soils are gener-ally subject to more stresses and productivityconstraints than is characteristic of the major UScrop-growing areas (Pimentel, 1993; Buol, 1994).This provides strong evidence for a yield limitsignificantly lower than theoretical potential.

However, we might apply the logistic growth

principle in a more modest way by assuming thatit will ultimately be possible to triple yields inregions now producing at relatively low yields: forexample, to raise yields from 2 to 6 t/ha over aperiod of decades. If this were the case, an ade-quate supply of food would be available for adoubled population with about a 50% per capitaconsumption increase. It turns out, however, thatthere are very significant differences in the growthpaths which are necessary to achieve this goal,depending on whether we assume an exponentialor a logistic pattern of yield growth.

Consider Fig. 11, which shows a logistic curvefitted to cereal yield data for Argentina2, using

1 Data for all figures and tables was downloaded from theUN Food and Agriculture Organization World Wide Web sitehttp://www.fao.org

2 Argentina is chosen as an example of a country whosecurrent cereal yields are close to the world average of about2.8 t/Ha.


Fig. 2. Total cereal yields for Argentina, 1961–1995.

FAO data for the period 1961–1995, and assum-ing an ultimate limit of three times current yieldlevels. The upper curve shows the logistic growthpath which would have been required to achieveclose to a tripling of yields over present levels bythe mid-twenty-first century. The fitted logisticcurve indicates a significantly slower growth path,requiring over 100 years to triple yields, butachieving about a doubling of yields over 1995levels by 2050. Another way of looking at thesame data is to fit an exponential and a logistictrend line (Fig. 2): the two trends are almostindistinguishable through the year 2000, but di-verge dramatically by 2050.

For most countries, current rates of yieldgrowth, if sustained in an exponential pattern,would be sufficient to triple yields by the mid-twenty-first century. On the other hand, a logisticpattern would imply that current growth rates ofaround 4% would be needed to achieve a tripling

over this time period. This is about doublethe observed yield growth rates of 1961–1995(Table 1).

Do we believe, then, in an exponential or alogistic pattern for yield growth? The answer tothis question will largely determine our degree ofoptimism about regional and planetary carryingcapacity. Most studies concur that there is limitedscope for expansion of land area cultivated—fu-ture growth in output must come mainly fromimproved yields (Crosson and Anderson, 1994).3

The logistic patterns which fit observed trendsgenerally indicate a potential for doubling, ratherthan tripling, yields over the next 50 years. Thisleaves little margin for error throughout most of

3 See Harris (1996) for a discussion of the relative contribu-tions of land expansion and yield growth to future agriculturalproduction growth.


Table 1Growth rates

Logistic growth rate needed toExponential growth rate needed toAverage yield growth rate (loga-triple yields by 2060, 1961–2060rithmic), 1961–1995 triple yieldsa by 2060, 1961–2060

2.10 1.84 4.22Argentina (%)1.66Brazil (%) 3.651.82

Canada (%) 2.97 2.09 4.982.46China, mainland 6.014.01

(%)1.631.47 2.95China, Taiwan

(%)Egypt (%) 4.202.16 1.84

3.07 2.17France (%) 5.191.91India (%) 4.432.392.04Indonesia (%) 4.812.68

Kenya (%) 1.22 1.44 2.954.652.36 1.98Mexico (%)

1.80USA (%) 1.80 4.08Developed all 3.971.75 1.76

(%)4.512.47 1.94Developing all

(%)

a Both the exponential and the logistic growth rates were calculated using the yields for 1961 as a starting point.

the developing world, where population is ex-pected to double during this period. If we examineactual yield trends, we will find further reason forskepticism concerning the optimistic, exponential-growth projections.

3. Yield trends for major crops: maize, wheat,and rice

Overall, world cereal yield growth rates havedeclined during the period since 1961 (Fig. 34).This, of course, is generally consistent with alogistic rather than exponential growth path. Butwe can also discern a difference in the patterns ofyield growth between presently developed andpresently developing nations, possibly suggestingthat the two groups of nations may be in differentregions of a logistic growth curve. The pattern oftotal cereal yields for developed and developingnations from 1961 to 1995 is shown in Fig. 4.Yield growth rates in developed nations have

clearly slowed, and an oscillating or ‘sawtooth’pattern appears to have developed since the mid-1980s. Average yields in the developing nations,meanwhile, have grown steadily over the full pe-riod.5 We could hypothesize that developing na-tions are on the earlier portion of a logistic curve,and developed nations on the later portion. Fromthis perspective, it appears that there is about a15-year time-lag, that is to say, developing nationsare about 15 years behind developed nations inachieving a particular yield level. This would alsolead us to expect that within the next 15 years thedeveloping nations would start to experience the

5 In a comment on an earlier draft of this paper, FraserSmith noted that the greater variability apparent in the devel-oped-country yield trend may be partly due to the greaterreliability of output data for these countries, while samplingerror in the developing countries may cause an artificialsmoothing of their yield trend. Given the broad-gauge estima-tion methods used by the FAO for many developing countries,this is quite likely. But, as Fraser Smith also noted, theoscillating pattern of a logistic yield trend approaching carry-ing capacity is a phenomenon well known in populationbiology. See further comment in Appendix A of this paper.4 Fig. 3 is from Harris (1996).


Fig. 3. World cereal yield growth rates, 1961–1993, using a 5-year moving average.

slowing of yield growth rates which is alreadyapparent in developed nation agriculture.

The suggestion of a logistic pattern can be seenmore clearly by examining the yield growthrecords for three major cereal crops: maize, wheatand rice. Fig. 5 shows the secular yield growthpattern for maize in developed and developingnations. The rate of growth has apparently slowedalmost to a standstill for developed nations sinceabout 1980. Since 1980, we note no net increase inyields, but a pronounced yield variability. This isconsistent with the hypothesis that yields in devel-oped nations have reached upper limits, and thata particular year’s yield is now determined pri-marily by weather or other external variables. Arecent study by Naylor et al. (1997) providesstatistical support for this empirical observation;they also suggest that high-yield cultivars are es-pecially susceptible to weather-related yieldvariations.

Developing nations’ maize yields are steadilyincreasing, and currently less than half developednation levels.6 This clearly leaves considerablemargin for further growth in developing nations,but in the range of doubling rather than triplingyields. To do much better, we would have to shareWaggoner’s optimistic view that not only canyields continue to rise significantly in developednations, reversing the pattern of recent years, butalso that these gains can be transmitted to devel-oping nations.

6 The maize yield pattern clearly shows a smoother trend indeveloping nations, probably partly due to data estimationtechniques as noted in the previous footnote. But the devel-oped nation pattern of variability after 1980 is strongly consis-tent with the hypothesis of oscillations around a stable mean(in this case, about 6 t/ha). See Appendix A for furtherdiscussion of this point.


Fig. 4. Total cereal yields developed and developing countries, 1961–1995.

Fig. 6 shows the story for wheat yields. Herealso developed country yields seem to show aslowdown in growth, with an actual decline overthe last 5 years. But in the case of wheat, develop-ing countries also appear to show some slow-down, and their achieved levels as of 1995 areapproximately equal to those of developed na-tions. This leaves less margin for growth in eitherdeveloping or developed nations. These trends, ofcourse, could be reversed, but the dose of opti-mism necessary to project this is larger for wheatthan for maize.

In Asia, where rice is of particular importance,the pattern of yield growth for major nations isshown in Fig. 7. Japanese yields have been stag-nant for the last quarter-century, at about 6 t/ha.7

Taiwan and China have increased yield steadily,and are now approaching the Japanese level. In-donesian rice yields have grown more slowly,reaching about two-thirds of Japanese levels,while Indian rice yields are currently about halfthe Japanese benchmark. These data are sugges-tive of a ceiling on yields, represented by intensiveJapanese cultivation, with other major producersapproaching this ceiling. This is consistent withagronomic research indicating that the climate-ad-justed yield potential for rice in Asia is 8.6 t/ha,but that production of rice on a commercial scalerarely exceeds 80% of theoretical potential (Cass-man and Harwood, 1995).

Thus for all three of these major cereal crops,the observed pattern of yield growth is suggestiveof a logistic rather than an exponential trend. InAppendix A, we compare the statistical fit ofexponential and logistic models to the data. Theresults, while not conclusive, generally support the

7 Like developed country maize yields, Japanese rice yieldsappear to show a pattern of oscillation around a stable meanof about 6 t/ha since about 1970.


Fig. 5. Developed and developing countries maize yields, 1961–1995.

logistic hypothesis. The observed patterns of yieldincrease are similar to the patterns which popula-tion biologists have identified as typical of popu-lations reaching carrying capacity limits.8 As wehave noted, such limits are generally absent fromeconomic models. This suggests that economicmodels used for agricultural projections, such asthose by the FAO, IFPRI, and the World Bankreferred to above, need to be modified to takeaccount of biophysical limits.

If indeed the developed nations are in the topportion of a logistic curve, we cannot expectdramatic further gains in yields for these countries(with the possible exception of the former SovietUnion, where gross inefficiencies have reducedagricultural productivity). We can also expect thatthe developing nations which have done well re-cently in raising yields may have trouble sustain-ing their recent growth rates (China in particular

bears watching in this regard). Two ‘yield gaps’still remain to be exploited: the difference betweentheoretically achievable and achieved levels inhigh-yield countries, and the difference betweenpresently high-yield and presently low-yield coun-tries. But the size of these gaps may be limited,and for some regions these limits may be quitestringent.

4. Some regional issues

Our earlier paper (Harris, 1996) projected thatan approximate doubling of cereal yields through-out the developing world would be necessary forself-sufficiency in the year 2025, with an estimated2025 world population of 8.3 billion, including 7billion in the presently developing world. Theseprojections are roughly similar to estimates ofslightly more than doubled cereal demand in de-veloping nations by 2025 (Alexandratos and de8 See, for example, Begon et al., 1996.


Fig. 6. Developed and developing countries wheat yields, 1961–1995.

Haen, 1995). A logistic yield growth trend isprobably consistent with a doubling of developingnation yields, but leaves little further room forgrowth, a major concern if we extend our projec-tion period to 2050, when current UN medianprojections indicate a world population between 9and 10 billion, with over 8 billion in the currentlydeveloping nations (United Nations, 1997). Im-ports from developed nations could serve as asafety valve; almost all projections show net im-ports by the developing world at least doubling by2025 (Islam, 1995). But of course in order tosupply these imports, the developed world mustincrease production, boosting either yields or areacultivated. There are also significant economicimplications to permanent import dependence, es-pecially for the least developed nations. The logis-tic analysis suggests that the world foodsupply/demand balance could tighten significantlyduring this period. While there is no necessaryindication of massive shortfalls, even a modest

tightening can drive prices up sharply (we haveseen a recent example of this in the 1995–1996cereal price spike).

Moving from a global to a regional perspective,we find wide differences both in populationgrowth rates and in agricultural yield trends formajor crops. It is important, therefore, to con-sider the picture for large regions. In each regionthe nature of the problem, and the probable con-straints on carrying capacity, are different.

In Africa, yields are generally low, and rates ofyield growth are also low. FAO data show cerealyields in Africa barely increasing over the last 15years (FAO, 1994). The pattern of Kenyan maizeyields shown in Fig. 8 is typical of the virtuallyflat yield growth record for much of Africa. Thisleaves a large theoretical yield gap to be exploited.However, rates of population growth in Africa arethe highest in the world, with a near-doublingover 1995 population levels projected by 2025,and close to a tripling likely by 2050 (Population


Fig. 7. Selected country paddy rice yields, 1961–1995.

Reference Bureau, 1998; United Nations, 1997).Concerns here are centered on the institutionaldifficulties of reversing a long-term low yieldgrowth trend, as well as on water limits through-out much of the region. For self-sufficiency,Africa would need to nearly triple yields by 2025,and quadruple them by 2050, assuming a 0.5%rate of per capita consumption growth to over-come current nutritional insufficiency and a 1%per annum rate of cultivated land expansion(Harris, 1996). If this yield growth is not feasible,the gap will have to be filled by a massive increasein imports, which many poorer African nationscan ill afford.

In east and southeast Asia, as we have noted,the importance of rice and the relatively narrowexploitable gap in rice yields suggest limits topopulation carrying capacity; rates of populationgrowth are slower here, but at least 500 millionpeople will be added by 2025 (Population Refer-

ence Bureau, 1998). Chinese rice yields, as wehave noted, are approaching the Japanese level,which may represent an achievable maximum.Maize yields in China have risen steadily, ap-proaching US levels (Fig. 8); wheat yields havenow surpassed US levels (Fig. 9). Wheat yieldscan rise significantly higher under favorable rain-fall conditions, as shown by French and Britishyields of around 6–7 t/ha, but water supply is amajor constraint on rainfed wheat.

In south Asia, a steady rate of growth in Indianwheat yields (Fig. 9) has also equalled the USyield level; water constraints here will certainly bea major issue for future yield growth. Rice yieldscould in theory double to Japanese levels (Fig. 7);again the water constraint is important. SouthAsia will certainly need to double overall cerealyields to accommodate population and demandgrowth; sustaining the steady yield growth to datewill require extensive investment in irrigation, and


Fig. 8. Selected country maize yields, 1961–1995.

will pose problems of absolute water limits andcompetition between agriculture, industry, andurban areas for available supplies.9

In Latin America, average cereal yields arenow around 2.5 t (Fig. 10). Population and de-mand growth through 2025 could be accommo-dated by raising these averages to the current USaverage level of 4–5 t/ha. To achieve this wouldrequire the appropriate inputs of water and nutri-ents as well as institutional infrastructure. This ispossible, and consistent with the logistic patternof yield growth shown above for Argentina inFig. 1. But it requires a degree of agronomic andinstitutional optimism to project that this will infact be achieved and sustained throughout theregion.

9 Ruttan (1993) notes the increasing importance of waterresource constraints on agricultural production in Asia andelsewhere.

5. Conclusions

There are strong indications in the data oncereal yields that a logistic pattern includingsome upper limit should be favored over an ex-ponential growth assumption. A logistic growthpattern is consistent with a rough doubling ofyields throughout the developed world. Thiswould be just about enough, assuming someland expansion and some increase in importsfrom presently developed nations, to meet theprojected consumption needs of the developingworld in 2025. But several strong caveats applyas follows.� The rice yield gap for major countries does not

appear to be great, putting projections of adoubling of rice yields in doubt.

� The yield gap for rainfed wheat is also small,and the potential for duplicating the perfor-mance of current high-yield nations is limitedby weather conditions.


Fig. 9. Selected country wheat yields, 1961–1995.

� There is a possibility, but no guarantee, offurther growth in yields in the already high-yield nations. The yield patterns for intensiverice agriculture in Japan and intensive maizecultivation in the US suggest that some practi-cal yield ceilings exist not far above presentlevels.

� Water supplies represent a significant con-straint on yield growth throughout much ofAfrica and Asia.

� In a context of limited rather than unlimitedyield growth, productivity losses to erosion andsoil degradation bulk larger. The pattern of thepast 35 years, where such losses are over-whelmed by steady and rapid yield growth, nolonger applies.

� The environmental damages associated withintensive agriculture—including soil degrada-tion, fertilizer and pesticide run-off, water pol-lution and overdraft—all become moredifficult to manage when demand pressures

militate against such measures as crop rotation,fallowing, and low-input techniques.

� The ecosystem damage and biodiversity lossassociated with the spread of cultivation ontomarginal lands will also be more prevalent in ahigh-demand scenario.This perspective is very different from the sup-

ply-side optimism characteristic of most econo-metric projection models. It implies a worldapproaching absolute carrying capacity limitsover the next few decades, with binding regionalconstraints, even if everything goes right in termsof agricultural investment and institutional in-frastructure. It is also important to recall that thegoal of feeding 8 billion people must not merelybe achieved, but must be sustained, taking intoaccount such cumulative problems as soil erosion,agriculture’s direct and indirect contribution togreenhouse gas accumulation, and depletion ofgroundwater supplies.

If we accept this rough indication of carrying


Fig. 10. Total cereal yields for selected countries, 1961–1995.

capacity limits, patterns of population growthbecome critical. Projections, of course, are onlyprojections; most median estimates of world pop-ulation for the year 2025 are around 8 billion, butwith a range of about 1 billion between the lowestand highest projections for that year. The dispar-ity becomes dramatically larger for 2050; the mostrecent UN series shows a low estimate for 2050 of7.7 billion (implying that world populationgrowth will have ceased and a small decline begunaround 2030), and a high estimate of 11.2 billion(United Nations, 1997; Population Reference Bu-reau, 1998).

Our analysis of yield trends implies that whilethe lower figure is within carrying capacity subjectto reasonable assumptions, the higher figure isnot. Even the UN median estimate of 9.4 billionfor 2050 seriously strains carrying capacity on theassumption of logistical trends in grain yields.Chen and Kates, who favor a higher-range popu-lation estimate, suggest a ‘normative’ requirement

of a three- to fourfold increase in food suppliesfor nutritional security in 2060 (Chen and Kates,1994). This is clearly outside the range of reason-able expectations given logistical yield trends.

Some analysts, such as Seckler and Cox (1994)view the ‘low’ series as the most likely long-termestimate of global population trends, given ob-served patterns of fertility decline. If this is borneout, the prospects for maintaining adequate foodsupplies would clearly be much brighter. But aswe move towards the median or high populationgrowth variants, the likelihood of greater environ-mental damage and biodiversity loss, as well asthe possibility of serious food shortages, becomesmuch greater.

The other important demand-side variable isper capita consumption. Economic growth hasgenerally been associated with increased demandfor feedgrains, which greatly increases the overallincome elasticity of demand for grains. The de-mand projections we have discussed assume a


modest, 0.5% per annum increase in demand forcereals throughout the developing world. A pat-tern of steady increase in meat and dairy productconsumption could easily double this estimatedgrowth rate; (the recent trends in Chinese directand indirect cereal consumption bear this out(Brown, 1995)). Thus a lower population projec-tion could easily be offset by more rapid percapita demand growth.

The lack of upper limits in most economicmodels of agricultural growth leads to an exces-sive emphasis on expansion of production, and aninsufficient consideration of environmental con-straints and the need for population limits. Alogistical growth model, for which there is strongsupportive evidence, should lead us to focus in-stead on environmentally sustainable productiontechniques, efficiency in consumption, and mea-sures to limit population growth.

Acknowledgements

We would like to acknowledge the support ofthe Tufts University Global Development andEnvironment Institute. The paper has benefittedfrom the comments of Fraser Smith and BobSteffen. Versions of this paper were presented atthe fourth biennial meeting of the InternationalSociety for Ecological Economics at Boston Uni-versity in August 1996, and at the seventh Inter-national Congress of Ecology at Florence, Italy inJuly 1998. Ideas presented in this paper weredeveloped in discussions with Bruce Hannon,Charles Hall, Kenneth Cassman, Gil Pontius, Pe-ter Rossett, Molly Anderson, Richard Wetzlerand Paul Kirshen and with participants in theKeystone Center Workshop on Critical Variablesand Long-Term Projections for SustainableGlobal Food Security in March 1997. None ofthese individuals or institutions bears any respon-sibility for any of the conclusions presented here,or for any errors or omissions in our analysis.

Appendix A. Statistical analysis of yield trends

We have performed an initial statistical test ofthe goodness of fit for exponential and logistic

models of yield trends. The results are shown inTable 2. The logistic fit is generally better than theexponential fit for all cereals together, as well asfor most crops and major regions. In many cases,the margin of superiority of logistic over exponen-tial fits, in terms of variance explained, is so slimthat it would be unjustifiable to draw any definiteconclusion. But for major crops in developednations, there is strong evidence of a logisticpattern.

The logistic model10 used for estimation is:

y=k/(1+exp(−r(x−h)))

where y is yield, x is year, k is logistic upperlimit, r is intrinsic growth rate, and h is year inwhich 50% of k is reached (inflection point oflogistic).

The goodness of fit of logistic and exponentialmodels is compared using the Akaike InformationCriterion (AIC).11 Both in terms of varianceexplained and AIC, the logistic model is superiorfor all developed nation crop yields. For devel-oping nations, there is generally little differ-ence between the two models in variance ex-plained; the AIC criterion supports the logisticmodel for wheat and all cereals together, while theexponential is marginally favored for maize andrice.

Fig. 11 shows the logistic fit for overall cerealyields in developed countries. There is a clearpattern of declining rates of yield growth throughthe whole period 1960–1995. Further, the pro-jected value of k, the logistic upper limit, is onlyslightly above present yield levels. For developingcountries (Fig. 12), a logistic fit would imply a

10 The use of the logistic model for analyzing populationgrowth in natural systems is discussed in Lotka (1956), Pielou(1969), Clark (1990), and Brown and Rothery (1993).

11 See Hilborn and Mangel, 1997, p. 160 for a discussion ofthe derivation and use of the AIC criterion. AIC is calculatedhere using Hongzhi’s proposed analog of the AIC for use withsum of squares. The formula is AIC= log(SSQk)+2k/n,where SSQk is the residual sum of squares for the model withk parameters and n is the number of data points. The modelwith the lowest AIC value is judged to be the model whichprovides the best fit to the underlying data.


Fig. 11. Cereal yields for developed countries, y= (3.575292)/(1+exp(− (0.06462311)*(x− (1963.22)))).

casts doubt on the applicability of the more opti-mistic exponential model.

Fig. 13 shows the logistic fit for maize in devel-oped nations, which is superior to the exponentialin terms of variance explained. Here we also seean apparent pattern of oscillation around thelogistic upper limit k, a characteristic often noted

yield ceiling about 50% above present levels. Anexponential fit, which is almost as good for devel-oping country data, would make goals of dou-bling or tripling yields quite feasible. It is theexperience of developed countries with decliningyield growth, rather than any statistical inferencefrom existing developing nation experience, which

Table 2Comparison of model results

Country designa- Variance explainedProduct designa- AIC comparison Best fit modeltion tion

ExponentialLogisticLogistic (%) Exponential(%)

−0.1012121 0.025717655 LogisticAll cerealsDeveloped coun- 93.258 89.699tries

Developing coun- Logistic−0.821113463−1.011765798.42199.108All cerealstries

85.964 1.015496753Maize Logistic80.797Developed coun- 0.9365222tries

−0.8001780 −0.802551359 ExponentialMaizeDeveloping coun- 98.663 98.483tries

Japan 0.902363908 LogisticPaddy rice 33.03442.933 0.8900355−0.533957998.260 Exponential−0.55646123398.394Developing coun- Paddy rice

triesLogisticWheat 92.830 89.868 −0.1824714 −0.089424399Developed coun-

triesWheat −0.439650517 Logistic98.363 97.315Developing coun- −0.5975463

tries0.043320328 LogisticWheatUSA 80.245 74.690 −0.0071496


Fig. 12. Cereal yields for developing countries, y= (4.384364)/(1+exp(− (0.04356363)*(x− (1985.791)))).

Fig. 13. Maize yields for developed countries, y= (6.728744)/(1+exp(− (0.08102871)*(x− (196.277)))).

Fig. 14. Wheat yields for the United States, y= (2.666169)/(1+exp(− (0.0737402)*(x− (1955.385)))).


Fig. 15. Paddy rice yield for Japan, y= (6.048986)/(1+exp(− (0.1326752)*(x− (1950.622)))).

bling or tripling yields quite feasible. It is theexperience of developed countries with decliningyield growth, rather than any statistical inferencefrom existing developing nation experience, whichcasts doubt on the applicability of the more opti-mistic exponential model.

Fig. 13 shows the logistic fit for maize in devel-oped nations, which is superior to the exponentialin terms of variance explained. Here we also seean apparent pattern of oscillation around thelogistic upper limit k, a characteristic often notedin population dynamics models.12 The most likelypractical interpretation for this phenomenon isthat the technical possibilities for increasing yieldshave largely been exploited, leaving weather andother random factors as the remaining determi-nant of year-to-year yield levels. For maize indeveloping countries, the logistic and exponentialfits are essentially equally good, which could indi-cate an exponential growth pattern but is alsoconsistent with the hypothesis that these countriesare in the initial phase of a logistic pattern whiledeveloped nations have reached the later phase.

Fig. 14 indicates a similar pattern for US wheatyields. The logistic fit, which is better than theexponential by both variance explained and AICcriteria, indicates that a yield ceiling has essen-tially been reached, with oscillation around

12 Population oscillation around an upper logistic limit isdiscussed in May (1974, 1976), and Begon et al. (1996).

the logistical upper limit of about 2.6 t/ha evidentsince the 1980s. For wheat in developed nationsgenerally, the logistic model is a better fit than theexponential (Table 2), while for developing na-tions the logistic is also superior to the exponen-tial, reflecting a slight slowdown in yield growthsince the mid-1980s.

Fig. 15 clearly indicates oscillation around anupper limit of about 6–7 t/ha for Japanese riceyields. In this case, the pattern of oscillation hascontinued since the 1970s.13 The oscillation re-duces the proportion of variance explained byeither logistic or exponential models, but the lo-gistic variance explained is higher, and it isslightly better by the AIC criterion. For rice indeveloping countries as a whole, the exponentialis a slightly better fit using the AIC criterion, andthe logistic slightly better in terms of varianceexplained.

Overall, for developing countries the statisticalevidence for a logistic model is relatively weak,and for some crops the data lend themselvesequally well to an exponential model. However,there is compelling evidence from the data formajor crops in developed nations that logistictrends in yields can be expected at higher produc-tivity levels. Together with significant evidence of

13 The yield growth patterns already noted for China, Tai-wan, and Indonesia (Fig. 7) are consistent with a logistic upperlimit in this range.


oscillation around the apparent upper logistic limitk, this yield growth pattern clearly resembles thebehavior of populations approaching carrying ca-pacity, as described in many works on populationecology.14

While the theoretical logic underlying popula-tion ecology models necessarily differs from thetheoretical justification for crop yield limits, thereare some clear parallels. Food supply is a majordeterminant of population dynamics, as nutrientuptake is of crop yields. Predator/prey dynamicsaffect population oscillations in ecology, whileweather variation plays a similar role in cropyields. The analogy should not be overstretched,but the data reviewed here provide support for theproposition that food supply models should takeinto account the notion of carrying capacity limits.

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