uncertainty and sustainability in the management of rangelands

E C O L O G I C A L E C O N O M I C S 6 2 ( 2 0 0 7 ) 2 5 1 – 2 6 6

ava i l ab l e a t www.sc i enced i rec t . com

www.e l sev i e r. com/ l oca te /eco l econ

Uncertainty and sustainability in the managementof rangelands

Martin F. Quaasa,⁎, Stefan Baumgärtnerb, Christian Beckerb, Karin Franka, Birgit Müllera,1

aDepartment of Ecological Modelling, UFZ — Centre for Environmental Research Leipzig-Halle, Leipzig, GermanybResearch Center for Environmental Economics, University of Heidelberg, Germany

A R T I C L E I N F O

* Corresponding author. UFZ — Centre for EnvDepartmentof EcologicalModelling, POBox 50Germany.

E-mail address: [email protected] (M.F.1The contributions of all authors are equa

0921-8009/$ - see front matter © 2006 Elsevidoi:10.1016/j.ecolecon.2006.03.028

A B S T R A C T

Article history:Received 22 March 2005Received in revised form6 March 2006Accepted 16 March 2006Available online 10 July 2006

We analyze a dynamic and stochastic ecological–economic model of grazing managementin semi-arid rangelands. The ecosystem is driven by stochastic precipitation. A risk aversefarmer chooses a grazing management strategy under uncertainty such as to maximizeexpected utility from farming income. Grazing management strategies are rules aboutwhich share of the rangeland is given rest depending on the actual rainfall in that year. In afirst step we determine a myopic farmer's optimal grazing management strategy and showthat a risk averse farmer chooses a strategy such as to obtain insurance from the ecosystem:the optimal strategy reduces income variability, but yields less mean income than possible.In a second step we analyze the long-run ecological and economic impact of differentstrategies. We conclude that a myopic farmer, if he is sufficiently risk averse, will choose asustainable grazing management strategy, even if he does not take into account long-termecological and economic benefits of conservative strategies.

© 2006 Elsevier B.V. All rights reserved.

Keywords:Ecological–economic modelSemi-arid rangelandGrazing managementRisk aversionUncertaintySustainability

JEL classification:Q57, Q12, Q24

1. Introduction

There is a widely held belief that individual myopic optimiza-tion is at odds with long-term sustainability of an ecological–economic system. In this paper, we want to take a fresh look atthis position. We show that for typical ecosystems and underplausible and standard assumptions about individual decisionmaking, myopic optimization may lead to sustainable out-comes. In particular, in order to explain the sustainable use ofecosystems, it is not necessary to assume preferences for sus-tainability— or any special concern for the distant future— on

ironmental Research,0135, D-04301 Leipzig,

Quaas).lly important.

er B.V. All rights reserved

the part of the decision maker; it suffices to assume that amyopic decision maker is sufficiently risk averse.

The ecological–economic systemunder study here is grazinginsemi-arid rangelands. Semi-arid regionscoverone thirdof theEarth's land surface. They are characterized by low and highlyvariable precipitation. Their utilization in livestock farmingprovides the livelihood for a large part of the local populations.Yet, over-utilization and non-adapted grazing strategies lead toenvironmental problems such as desertification. Grazing insemi-arid rangelands is a prime object of study for ecologicaleconomics, as the ecological and economic systems are tightlycoupled (e.g. Beukes et al., 2002;Heady, 1999; Janssen et al., 2004;Perrings, 1997; Perrings and Walker, 1997, 2004; Westoby et al.,1989). The grass biomass is directly used as forage for livestock,which is the main source of income; and the grazing pressurefrom livestock farming directly influences the ecological dy-namics. The crucial link is the grazing management.

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http://dx.doi.org/10.1016/j.ecolecon.2006.03.028

252 E C O L O G I C A L E C O N O M I C S 6 2 ( 2 0 0 7 ) 2 5 1 – 2 6 6

The ecological dynamics, and thus, a farmer's income,essentially depend on the low and highly variable rainfall. Thechoice of a properly adapted grazing management strategy iscrucial in two respects: first, tomaintain the rangeland systemas an income base, that is, to prevent desertification; andsecond, to smooth out income fluctuations, in particular, toavoid high losses in the face of droughts.

Assuming that the farmer is non-satiated in income and riskaverse,we analyze the choice of a grazingmanagement strategyfrom two perspectives. In a first step we determine a myopicfarmer's optimal grazingmanagement strategy.We show that arisk averse farmer chooses a strategy in order to obtain‘insurance’ from the ecosystem (Baumgärtner and Quaas,2005). That is, the optimal strategy reduces income variability,but yields less mean income than possible. In a second step weanalyze the long-term ecological and economic impact ofdifferent strategies. We conclude that the more risk averse amyopic farmer is, the more conservative is his optimal grazingmanagement strategy. If he is sufficiently risk averse, theoptimal strategy is conservative enough to be sustainable.

Following the literature on grazing management underuncertainty, we analyze the choice of a stocking rate of livestock,as this is the most important aspect of rangeland management(e.g. HeinandWeikard, 2004; KarpandPope, 1984;McArthur andDillon, 1971; Perrings, 1997; Rodriguez andTaylor, 1988; Torell etal., 1991; Westoby et al., 1989). The innovative analyticalapproach taken here is to consider the choice of a grazingmanagement strategy, which is a rule about the stocking rate toapply in any given year depending on the rainfall in that year.This is inspired by empirical observations in Southern Africa.Rule-based grazing management has the twofold advantagethat a farmer has to make a choice (concerning the rule) onlyonce, and yet, keeps a certain flexibility and scope for adaptivemanagement (concerning the stocking rate). The flexibility thusobtained is the decisive advantage of choosing a constant ruleover choosing a constant stocking rate.

The paper is organized as follows. In Section 2, we discussgrazingmanagement in semi-arid rangelands inmore detail anddescribe one particular ‘good practice’—example: the GamisFarm,Namibia. InSection3,wedevelopadynamicandstochasticecological–economicmodel,whichcaptures theessential aspectsand principles of grazing management in semi-arid rangelands,and features the key aspect of theGamis-strategy. Our results arepresented inSection4,withall derivationsandproofsgiven in theAppendix. Section 5 concludes.

2. Grazing management in semi-aridrangelands: the Gamis Farm, Namibia

Thedynamicsofecosystems insemi-arid regionsareessentiallydriven by low and highly variable precipitation (Behnke et al.,1993; Sullivan and Rhode, 2002; Westoby et al., 1989).2 Sustain-able economic use of these ecosystems requires an adequateadaption to this environment. The only sensible economic use,

2 Another important driver of ecological dynamics in semi-aridrangelands is the stochastic occurrence of fire (Janssen et al., 2004;Perrings andWalker, 1997, 2004). In our case, fire plays only aminorrole but the stochasticity of rainfall is crucial (Müller et al., in press).

which is indeed predominant (Mendelsohn et al., 2002), is byextensive livestock farming. However, over-utilization andinadequate management lead to pasture degradation anddesertification. Rangeland scientists have proposed differenttypes of grazing management strategies in order to solve theseproblems. A low constant stocking rate was recommended byLamprey (1983) and Dean and MacDonald (1994), who assumedthat grazing pressure is the main driving force for vegetationchange and that rangeland systems reach an equilibrium state.Other authors considered the highly variable rainfall to be themajor driving force and claimed that grazing has only a mar-ginal influence on vegetation dynamics (Behnke et al., 1993;Scoones, 1994; Sandford, 1994; Westoby et al., 1989). They re-commended an ‘opportunistic’ strategy which matches thestocking rate with the available forage in every year. Thus, thestocking rate should behigh inyearswith sufficient rainfall, andlow when there is little forage in dry years (Beukes et al., 2002:238). Recent studies have shown that both grazing and variablerainfall are essential for the vegetation dynamics on differenttemporal and spatial scales (Cowling, 2000; Briske et al., 2003;Fuhlendorf and Engle, 2001; Illius and O'Connor, 1999, 2000;Vetter, 2005).

One example of a sophisticated and particularly successfulgrazing management system has been employed for 40 yearsat the Gamis Farm, Namibia (Müller et al., in press; Stephanet al., 1996, 1998a,b). The Gamis Farm is located 250 kmsouthwest of Windhoek in Namibia (2405′ S 1630′ E) close tothe Naukluft mountains at an altitude of 1250 m. The climateof this arid region is characterized by low mean annualprecipitation (177 mm/y) and high variability (variationcoefficient: 56%). The vegetation type is dwarf shrub savanna(Giess, 1998); the grass layer is dominated by the perennialgrasses Stipagrostis uniplumis, Eragrostis nindensis and Triraphisramosissima (Maurer, 1995).

Karakul sheep (race Swakara) are bred on an area of30,000 ha. The primary source of revenue is from the sale oflamb pelts. Additionally, the wool of the sheep is sold. In goodyears, up to 3000 sheep are kept on the farm. An adaptivegrazing management strategy is employed to cope with thevariability in forage. The basis of the strategy is a rotationalgrazing scheme: the pasture land is divided into 98 paddocks,each of which is grazed for a short period (about 14 days) untilthe palatable biomass on that paddock is used up completely,and then is rested for aminimumof 2months. This systemputshigh pressure on the vegetation for a short time to preventselective grazing (Batabyal andBeladi, 2002; Batabyal et al., 2001;Heady, 1999). While such a rotational grazing scheme is fairlystandard throughout semi-arid regions, the farmer on theGamis Farm has introduced an additional resting: in yearswith sufficient precipitation one third of the paddocks are givena rest during the growth period (September–May). In years, withinsufficient rainfall this rest period is reduced or completelyomitted. Once a year, at the end of the rainy season (April), thefarmer determines — based on actual rainfall and availableforage — how many paddocks will be rested and, thus, howmany lambs can be reared. This strategy is a particular exampleof what has been called ‘rotational resting’ (Heady, 1970, 1999;Stuth and Maraschin, 2000; Quirk, 2002).

The grazing management system employed at the GamisFarm has been successful over decades, both in ecological and

253E C O L O G I C A L E C O N O M I C S 6 2 ( 2 0 0 7 ) 2 5 1 – 2 6 6

economic terms. It, therefore, represents a model for com-mercial farming in semi-arid rangelands.

3. The model

Our analysis is based on an integrated dynamic and stochasticecological–economic model, which captures essential aspectsand principles of grazing management in semi-arid regions. Itrepresents a dynamic ecosystem, which is driven by stochas-tic precipitation, and a risk averse farmer, who rationallychooses a grazing management strategy under uncertainty.

3.1. Precipitation

Uncertainty is introduced into the model by the stochasticityof rainfall, which is assumed to be an independent andidentically distributed (iid) random variable. For semi-aridareas, a log-normal distribution of rainfall r(t) is an adequatedescription (Sandford, 1982).3 The log-normal distribution,with probability density function f(r) (Eq. (A.25)), is determinedby the mean μr and standard deviation σr of precipitation.Here, we measure precipitation in terms of ‘ecologicallyeffective rain events’, i.e. the number of rain events duringrainy season with a sufficient amount of rainfall to beecologically productive (Müller et al., in press).

3.2. Grazing management strategies

The farm is divided into a number I∈ IN of identical paddocks,numbered by i∈ {1, …, I}. In modeling grazing managementstrategies, we focus on the aspect of additional resting duringthe growth period, which is the innovative element in theGamis grazing system. That is, we analyze rotational resting ofpaddocks from year to year, but do not explicitly considerrotational grazing during the year (cf. Section 2). The strategyis applied in each year, after observing the actual rainfall at theend of the rainy season. Its key feature is that in dry years allpaddocks are used, while in years with sufficient rainfall a pre-specified fraction of paddocks is rested. Whether resting takesplace, and to what extent, are the defining elements of whatwe call the farmer's grazing management strategy:

Definition 1. A grazing management strategy (α, r_ ) is a rule of howmany paddocks are not grazed in a particular year given theactual rainfall in that year, where α∈ [0, 1] is the fraction ofpaddocks rested if rainfall exceeds the threshold value r_ ∈ [0,∞).4

Thus, when deciding on the grazing management strategy,the farmer decides on two variables: the rain threshold r_ andthe fraction α of rested paddocks. While the rule is constant(i.e. α=const., r_ =const.) its application may yield a differentstocking with livestock in any given year depending on actualrainfall in that year.

In the resource economics literature, this type of strategy iscalled ‘proportional threshold harvesting’ (Lande et al., 2003).

3 While the distribution of rainfall r(t) is exogenous, all otherrandom variables in the model follow an induced distribution.4 We assume that the number I of paddocks is so large that we

can treat α as a real number.

This is a form of adaptive management: the (constant) ruleadapts the fraction of fallow paddocks and the number oflivestock kept on the farm as actual rainfall changes. Note thatthe ‘opportunistic’ strategy (e.g. Beukes et al., 2002: 238) is thespecial case without resting, i.e. α=0.

3.3. Ecosystem dynamics

Both the stochastic rainfall and grazing pressure are majordeterminants of the ecological dynamics. Following Stephanet al. (1998a), we consider two quantities to describe the stateof the vegetation in each paddock i at time t: the greenbiomass Gi(t) and the reserve biomass Ri(t) of a representativegrass species,5 both of which are random variables, since theydepend on the random variable rainfall. The green biomasscaptures all photosynthetic (‘green’) parts of the plants, whilethe reserve biomass captures the non-photosynthetic reserveorgans (‘brown’ parts) of the plants below or above ground(Noy-Meir, 1982). The green biomass grows during the growingseason in each year and dies almost completely in the courseof the dry season. The amount Gi(t) of green biomass availableon paddock i in year t after the end of the growing seasondepends on rainfall r(t) in the current year, on the reservebiomass Ri(t) on that paddock, and on a growth parameter wG:

GiðtÞ ¼ wGd rðtÞd RiðtÞ: ð1ÞAs the green biomass in the current year does not directly

depend on the green biomass in past years, it is a flow variablerather than a stock.

In contrast, the reserve biomass Ri(t) on paddock i in year tis a stock variable. That is, the reserve biomass parts of thegrass survive several years (‘perennial grass’). Thereby, thedynamics of the vegetation is not only influenced by thecurrent precipitation, but also depends on the precipitation ofpreceding years (O'Connor and Everson, 1998). Growth of thereserve biomass from the current year to the next one is

Riðtþ 1Þ−RiðtÞ ¼ −dd RiðtÞd 1þ RiðtÞK

� �

þwRd ð1−cd xiðtÞÞd GiðtÞd 1−RiðtÞK

� �; ð2Þ

wherewR is a growth parameter and d is a constant death rateof the reserve biomass, which we assume to be sufficientlysmall, i.e. d<wR wG μr. A density dependence of reservebiomass growth is captured by the factors containing theparameter K: The higher the reserve biomass on paddock i, theslower it grows. The status variable xi captures the impact ofgrazing on the reserve biomass of paddock i. If paddock i isgrazed in year t, we set xi(t)=1, if it is rested, we set xi(t)=0. Theparameter c (with 0≤c≤1) describes the amount by whichreserve biomass growth is reduced due to grazing pressure.For simplicity, we assume that the initial (t=1) stock of reservebiomass of all paddocks is equal,

Rið1Þ ¼ R for all i ¼ 1; N ; I: ð3Þ

5 We assume that selective grazing is completely prevented, i.e.there is no competitive disadvantage for more palatable grasses(see e.g. Beukes et al., 2002). Hence, we consider a single,representative species of grass.


3.4. Livestock and income

As for the dynamics of livestock, the herd size S(t), that can bekept on the farm at time t, is limited by total available forage.We normalize the units of green biomass in such a way thatone unit of green biomass equals the need of one livestockunit per year. Thus, total available green biomass on the farm,PI

i¼1 GiðtÞ, determines the ‘carrying capacity’, i.e. the maxi-

mum number of livestock that can be held on the farm in theperiod under consideration.6 In general, the farmer will notstock up to this carrying capacity in every year. Rather, theherd size kept on the farm in period t is given by

SðtÞ ¼XIi¼1

xiðtÞd GiðtÞ: ð4Þ

That is, the herd size in year t is determined by the totalgreen biomass of the paddocks used for grazing (i.e., notrested) in that year. For the sake of the analysis, we assumethat the farmer annually rents his livestock on a perfect rentalmarket for livestock.7 This allows the farmer to exactly adaptthe actual herd size to the available forage and to his chosengrazing management strategy.8

The herd size S(t) kept on the farm in year t determines thefarmer's income y(t). We assume that the quantity ofmarketable products from livestock, e.g. lamb furs and wool,is proportional to the herd size. Normalizing product units inan appropriate way, the numerical value of output equalslivestock S(t). The farmer sells his products on a world marketat a given price and takes the annual rental rate of livestock asgiven. The difference between the two is the net revenue perlivestock unit, p. Assuming that farming is profitable, i.e. p>0,the farmer's income y(t) is

yðtÞ ¼ pd SðtÞ: ð5ÞSince the herd size S(t) is a random variable, income y(t) is a

random variable, too.9 In order to simplify the notation in thesubsequent analysis, we normalize

puðwGd Id RÞ−1: ð6ÞThis means, from now on we measure net revenue per

livestock unit in units of total forage per unit of precipitation.As a result, income is measured in units of precipitation.

6 In contrast to the capacity limit K of reserve biomass, thecarrying capacity of livestock is not a constant, but it depends onrainfall and the stock of reserve biomass (cf. Eq. (1)), and,therefore, will change over time.7 If the farmer owns a constant herd of size S0, he would rent a

number S(t)−S0 if S(t)>S0 or rent out a number S0−S(t) if S(t)<S0.Without loss of generality, we set S0=0.8 Hence, the herd size S(t) does not follow its own dynamics, but

it is determined by precipitation and the chosen strategy.9 In our analysis, we neglect uncertainty of prices. Including a

price stochasticity uncorrelated to rainfall would not alter ourresults. Including a price stochasticity with a negative correlationto rainfall would most likely reinforce our central result that a riskaverse farmer chooses a conservative grazing managementstrategy, since high stocking rates in good rainy years becomeless valuable (as indicated by Hein and Weikard, 2004).

For the subsequent analysis of a myopic farmer's decision,first and second year incomeare of particular interest. Given theactual rainfall r(1) in the first grazing period, the initial reservebiomass (Eq. (3)) and a grazingmanagement rule (α, r_ ), the herdsize S(1) is determined by Eq. (4). Inserting Eq. (1) and usingAssumption (3), as well as normalization (6), the farmer'sincome y(1) in the first grazing period is given by Eq. (5) as

yð1Þ ¼ 1I

XIi¼1

xið1Þd rð1Þ

¼ rð1Þ if rð1Þ V rð1−aÞd rð1Þ if rð1Þ > r

:

� ð7Þ

Given the probability density distribution f(r) of rainfall, themean μy(1)(α, r

¯) and the standard deviation σy(1)(α, r_ ) of the

first period's income are (see Appendix A.1)

lyð1Þða; rÞ ¼ lr−aZ 1

rrf ðrÞdr ð8Þ

ryð1Þða; rÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2r þ 2alr

Z 1

rrf ðrÞdr−a2½ Z 1

rrf ðrÞdr�2−að2−aÞZ 1

rr2f ðrÞdr

vuut ; ð9Þ

where μr and σr are the mean and the standard deviation ofrainfall.

The model implies that resting in the first period has apositive impact on reserve biomass and, thus, on futureincome. In particular, if the farmer applies a grazing manage-ment strategy (α, r

¯) with α>0 and r

¯<∞, rather than full

stocking, he can gain an extra income in the second year.Given the actual rainfall r(1) in the first year, the additionalreserve biomass in the second year is (cf. Eqs. (1), (2) and (3))

DR ¼ wRdwGd Id Rd 1−RK

� �d rð1Þd 0 if rð1ÞV r

P

a if rð1Þ > rP

:

�ð10Þ

This additional reserve biomass gives rise to extra greenbiomass growth, and, hence, to additional income in thesecond year (cf. Eqs. (1), (4), (5) and (10)):

Dyð2Þ ¼ wGd rð2Þd 1 if rð2ÞV r1−a if rð2Þ > r d wRd 1−

RK

� �d rð1Þd 0 if rð1ÞV r

a if rð1Þ > r :��

ð11ÞThismeans, the reserve biomass can be used as a buffer: by

applying a grazing strategy with resting, the farmer can shiftincome to the next year. For a risk averse farmer, this extraincome is particularly valuable if the second year is a dry year.

3.5. Farmer's choice of grazing management strategy

We assume that the farmer's utility only depends on incomey, and that he is a non-satiated and risk averse expectedutility maximizer. Let

U ¼Xlt¼1

EtuðyðtÞÞð1þ dÞt−1 ð12Þ

be his von Neumann–Morgenstern intertemporal expectedutility function, where δ is the discount rate, the Bernoulliutility function u(·) is a strictly concave function of income y,

Fig. 1 A set of indifference curves of the risk averse farmer inthe mean–standard deviation space for log-normally dis-tributed incomes and constant relative risk aversion ρ=1.


and Et is the expectancy operator at time t. In particular, weemploy a utility function with constant relative risk aversion,

uðyÞ ¼ y1−q−11−q

; ð13Þ

where ρ>0 is the constant parameter which measures thedegree of relative risk aversion (Gollier, 2001).

The farmer will choose the grazing management strategywhich maximizes his von Neumann–Morgenstern intertem-poral expected utility function (Eq. (12)). The basic idea is toregard the choice of a grazing management strategy as thechoice of a ‘lottery’ (Baumgärtner and Quaas, 2005). Each pos-sible lottery is characterized by the probability distribution ofpay-off, where the pay-off is given by the farmer's income.Given the ecological dynamics, both themean income and thestandard deviation solely depend on the grazingmanagementstrategy applied. Thus, choosing a grazing managementstrategy implies choosing a particular distribution of income.

We assume that the farmer initially, i.e. at t=0 prior to the firstgrazing period, chooses a grazing management strategy (α, r

¯),

which is thenapplied in all subsequent years.When choosing thestrategy, the farmer does not knowwhich amount of rainfall willactually occur, but he knows the probability distribution ofrainfall. As a result, he knows the probability distribution of hisincome for any possible grazing management strategy. A far-sighted farmer would choose the grazing management strategythat maximizes his intertemporal utility (Eq. (12)), taking intoaccount the effect of the strategy on the ecosystem dynamics, asgiven by Eqs. (1) and (2). In particular, he would account for theeffect that resting improves the reserve biomass in the long run,compared to a strategy with full stocking. However, our aim is toshow that a sufficiently risk averse farmer will choose aconservative strategy, even if he does not consider the long-termbenefits. For this sake, we assume that the farmer ismyopicin the following sense (Kurz, 1987):

Definition 2. Amyopic farmerneglects the long-termeffects of hisgrazing management strategy on the ecosystem: (i) He assumesthat reserve biomass remains constant at the initial level R on allpaddocks, irrespective of the chosen strategy, with the exceptionthat (ii) he takes into account the extra income Δy (Eq. (11)) in ayear after resting.

This means, a myopic farmer bases his decision on a verylimited consideration of ecosystem dynamics: he only takes intoaccount the short-term buffering function of reserve biomass,while neglecting all long-term ecological impact of the grazingmanagement strategy chosen. Such a myopic farmer considershis income in year t≥2 to be

yðtÞ ¼rðtÞd 1 if rðtÞV r1−a if rðtÞ > r

� �

d 1þwRdwGd 1−RK

� �d rðt−1Þd 0 if rðt−1ÞV r

a if rðt−1Þ > r

� �� : ð14Þ

Since the myopic farmer neglects the long-term ecologicalimpact of his grazing strategy, the functional form of howannual income y(t) (Eq. (14)) depends on actual rainfall and onthe chosen strategy, remains constant over time. Further-more, since precipitation is independent and identicallydistributed in each year, and the strategy is constant, the

mean μy(t) and standard deviation σy(t) of the annual income y(t) for t≥2 are also constant over time.

In order to be able to express the expected instantaneousutility in any year t in terms of the mean and the standarddeviation of that year's income, we approximate the proba-bility density function of annual income by a log-normaldistribution with the same mean and standard deviation.Using the specification (Eq. (13)) of the Bernoulli utility func-tion u(y), expected instantaneous utility is given by thefollowing explicit expression (see Appendix A.2):

EuðyðtÞÞ ¼l1−qyðtÞð1þ r2yðtÞ=l

2yðtÞÞ−qð1−qÞ=2−1

1−q: ð15Þ

The indifference curves of the farmer's expected instanta-neous utility function can be drawn in the mean–standarddeviation space. Fig. 1 shows such a set of indifference curvesfor a given degree ρ of relative risk aversion.

The indifference curves are increasing and convex if thestandard deviation is sufficiently small compared to the mean,i.e. for (μy /σy)2>1+ρ (see Appendix A.3). The slope of theindifference curves is increasing in the degree of relative riskaversion ρ (see Appendix A.3). In particular, the indifferencecurves are horizontal lines for risk-neutral farmers, i.e. for ρ=0.

Formally, the decision problem to be solved by a myopicfarmer is to choose a grazingmanagement strategy (α, r

¯) such as

to maximize U (Eq. (12)) subject to Conditions (7), (14), (15). In thecontext of semi-arid rangelands, the growth rate of reservebiomass is small, i.e.wR≪1. In Appendix A.4we show that underthis condition the farmer's decision problem effectively becomes

maxða; r

PÞlyða; rPÞdð1þ r2yða; rPÞ=l2yða; rPÞÞ−q=2; ð16Þ

where the effectivemean and standard deviation of income are

lyða; rÞ ¼ ½lr−a Z 1

rrf ðrÞdr�d ½1þ aw

Z 1

rrfðrÞdr� ð17Þ

ryða; rÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2r þ 2alr

Z 1

rrfðrÞdr−a2

" Z 1

rrfðrÞdr

#2−að2−aÞ

Z 1

rr2f ðrÞdr

vuut

d

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 2aw

Z 1

rrfðrÞdr

s;

ð18Þ

with ω=wR ·wG · (1−R/K) / (1+δ). We analyze this decision prob-lem in the following.

Fig. 3 The set of feasible strategies is given by the whole areaαa [0, 1], r_ a [0,∞). The set of efficient strategies forparameters μr=1.2 and σr=0.7 is the curve.


4. Results

The analysis proceeds in three steps (Results 1, 2 and 3 below):First, we analyze the optimization problem of a risk aversemyopic farmerwho faces a trade-off between strategies whichyield a high mean income at a high standard deviation, andstrategies which yield a low mean income at a low standarddeviation. Second, we analyze the long-term consequences ofdifferent grazing management strategies on the ecological–economic system. In particular, we study how the long-termdevelopment of the mean reserve biomass and the meanincome depends on the strategy. Finally, we put the two partsof the analysis together and derive conclusions about how thelong-term sustainability of the short-term optimal strategydepends on the farmer's degree of risk aversion.

4.1. Feasible strategies and income possibility set

To start with, we define the income possibility set as the set ofall effective mean incomes and standard deviations of incomeμy(α, r

¯), σy(α, r

¯))∈ (0, ∞)× [0, ∞), which are attainable by applying

a feasible management rule (α, r¯)∈ [0,1]×[0, ∞). These are

given by Eqs. (17) and (18). Fig. 2 shows the income possibilityset for particular parameter values.

The figure provides one important observation: there existinefficient strategies, i.e. feasible strategies that yield thesame mean income, but with a higher standard deviation (or:the same standard deviation, but with a lower mean) thanothers. These strategies can be excluded from the set of stra-tegies from which the optimum is chosen by a risk averse andnon-satiated decision maker. In the following, we thus focuson the efficient strategies, which generate the income possibilityfrontier (Fig. 2, thick line):

Definition 3. The income possibility frontier is the set of expectedvalues μy and standard deviations σy of income for which thefollowing conditions hold:

1. (μy, σy) is in the income possibility set, i.e. it is feasible.2. There is no (μ′y, σ′y)≠ (μy, σy) in the income possibility set

with μ′y≥μy and σ′y≤σy.

Fig. 2 –The set of all means μy and standard deviations σy ofthe farmer's income y, each point denoting a separatestrategy, as well as the income possibility frontier (thick line).Parameter values are μr=1.2, σr=0.7 and ω=0.14.

The question at this point is, ‘What are the grazing manage-ment strategies (α, r

¯) that generate the income possibility

frontier?’We call these strategies efficient.

Lemma 1. The set of efficient strategies has the followingproperties.

• Each point on the income possibility frontier is generated byexactly one (efficient) strategy.

• There exists Ω⊆ [0, ∞), such that the set of efficient strategies isgiven by (α⁎( r

¯), r¯) with

a⁎ðrÞ ¼R1r rðr−rÞf ðrÞdrR1

r rðr−r=2Þf ðrÞdr for all r∈X: ð19Þ

• α⁎( r_ ) has the following properties:

a⁎ 0ð Þ ¼ 1; limrYl

a⁎ðrÞ ¼ 0; andda⁎ðrÞdr

< 0 for all r∈X:

Proof: see Appendix A.5.

Fig. 3 illustrates the lemma. Whereas the set of feasiblestrategies is the two-dimensional area bounded by r

¯=0, α=0,

α=1, the set of efficient strategies, as given by Eq. (19), is aone-dimensional curve. Thus, the efficient strategies aredescribed by only one parameter, r

¯, while the other

parameter α is determined by α=α⁎( r¯) (Eq. (19)). Alternatively,

the inverse function of Eq. (19) — which exists by Lemma 1 —specifies the efficient rain threshold r_ as a function of thefraction α of resting. The curve α⁎( r

¯) is downward sloping:

With a higher rain threshold r¯, i.e. if resting only takes place

in years with higher precipitation, the efficient fraction α⁎( r¯)

of rested paddocks is smaller. In other words, for efficientstrategies, a higher rain threshold r

¯does not only mean that

the condition for resting is less likely to be fulfilled, but alsothat a smaller fraction α⁎ of paddocks is rested if restingtakes place. Hence, if an efficient strategy is characterized bya smaller r

¯, and, consequently, by a larger α⁎( r

¯), we call it

more conservative.Knowledge of the efficient strategies allows us to charac-

terize the income possibility frontier, and to establish a

10 This is a sufficient conditionwhich is quite restrictive. A uniqueoptimum exists for a much larger range of parameter values.

Fig. 4 The optimum for a risk averse farmer (ρ=5.5, denotedby * ) and a risk-neutral farmer (ρ=0, denoted by +).


relationship between efficient grazingmanagement strategiesand the resulting means and standard deviations of income.

Lemma 2. The farmer's expected income in the first grazingperiod, μy(α, r_ ) (Eq. (17)), is increasing in r_ for all efficientstrategies:

dlyða⁎ðrÞ; rÞdr

> 0 for all r∈X:

The extreme strategies, r_ =0 and r_ →∞, lead to expected incomesof μy(α⁎(0), 0)=0 and lim

rYllyða⁎ ðrÞ; rÞ ¼ lr:


For all efficient strategies a higher rain threshold r_ forresting, i.e. a less conservative strategy, implies a highermeanincome. Whereas no resting, r_ →∞ (opportunistic strategy),leads to the maximum possible mean income of μr, the op-posite extreme strategy, r_ =0 (no grazing at all), leads to theminimum possible income of zero. Overall, a change in thegrazing management strategy affects both, the mean incomeand the standard deviation of income.

Lemma 3. The income possibility frontier has the following properties:

• The income possibility frontier has two corners:— The southwest corner is at σy=0 and μy=0. At this point, the

income possibility frontier is increasing with slope μr /σr.— The northeast corner is at σy=σr and μy=μr. At this point, the

income possibility frontier has a maximum and its slope is zero.• In between the two corners, the income possibility frontier is

increasing and located above the straight line from one corner tothe other. It is S-shaped, i.e. from southwest to northeast there isfirst a convex segment and then a concave segment.


Fig. 2 illustrates the lemma.Withno resting at all (northeastcorner of the income possibility frontier), the farmer obtainsthe highest possible mean income (μy=μr), but also faces thefull environmental risk (σy=σr). Conversely, with the mostconservative strategy, i.e. no grazing at all (southwest corner ofthe income possibility frontier), the farmer can completelyeliminate his income risk (σy=0), but also cannot expect anyincome (μy=0). The property, that the income possibilityfrontier is increasing, suggests that resting acts like aninsurance for the farmer. This means, by choosing a moreconservative grazing management strategy, the farmer cancontinuously decrease his risk (standard deviation) of income,but only at the price of a decreased mean income.

Thus, there is an insurance value associated with choos-ing a more conservative strategy (Baumgärtner and Quaas,2005).

4.2. Optimal myopic strategy

The optimal myopic strategy is obtained by solving Problem(16), and results from both the farmer's preferences (Fig. 1) andthe income possibility frontier (Fig. 2). In mean–standarddeviation space, it is determined by the mean μy⁎ and thestandard deviation σy⁎, at which the indifference curve istangential to the income possibility frontier (Fig. 4).

It turns out that theoptimal strategy is uniquelydetermined.

Lemma 4.

(i) If (μr /σr)2>1+ρ, the optimum (μy⁎, σy⁎) is unique.10

(ii) For ρ>0, the optimum is an interior solution with 0<μy⁎<μr and0<σy⁎<σr. For ρ=0, the optimum is a corner solutionwith μy⁎=μrand σy⁎=σr.


Theoptimalmyopic strategycrucially dependson thedegreeof risk aversion. In the particular case of a risk-neutral farmer(ρ=0), the strategy that yields themaximummean, irrespectiveof the standard deviation associated with it, is chosen. Theoptimal grazing management strategy of such a risk-neutralfarmer is the strategy without resting, i.e. with r_ =∞ (and,therefore, α=0). That is, he employs an opportunistic strategy.

If the farmer is risk averse, he faces a trade-off betweenexpected income and variability of the income, becausestrategies that yield a higher mean income also display ahigher variability of income. This leads to the following result,which is illustrated in Figs. 4 and 5.

Result 1. Aunique interior solution (α⁎(r_ ⁎), r_ ⁎) to the farmer's decisionProblem (16), if it exists (see Lemma 4), has the following properties:

(i) The more risk averse the farmer, the smaller are the mean μy⁎and the standard deviation μy⁎ of his income.

(ii) The more risk averse the farmer, the more conservative is hisgrazing management strategy:

dr⁎dq

< 0 andda⁎dq

> 0: ð20Þ


This means, a risk averse farmer chooses a grazingmanagement strategy such as to obtain insurance from theecosystem: by choosing a particular grazing managementstrategy the farmer will reduce his income risk, and carry theassociated opportunity costs in terms of mean incomeforegone (the ‘insurance premium’), to the extent that isoptimal according to his degree of risk aversion.

Fig. 5 The rain threshold r¯* of the optimal strategy as a

function of the farmer's degree of risk aversion ρ. Parametervalues are the same as in Fig. 2.

12 These steady-state mean values represent the trend of thestochastic dynamics, but not the purely random part of the


4.3. Long-term impact of grazing management strategies

To study the long-term ecological and economic impact of thegrazing management strategy chosen on the basis of myopicoptimization (Problem 16), we assume that the farmercontinues to apply this strategy in every subsequent period.We compute the resulting probability distribution of incomeand reserve biomass over several decades in the future. Thiscalculation covers all efficient strategies (α⁎( r_ ), r_ ). The resultsof the numerical computation11 are shown in Fig. 6, whichenables the comparison of the long-term impacts, both inecological and economic terms, of the different strategies thatare efficient from the viewpoint of a myopic farmer.

In this figure, the mean values μR(t) of reserve biomass andμy(t) of income at different times t are plotted against theefficient fraction α⁎( r_ ) of resting for different rain thresholdsr_ ∈Ω. The higher α⁎( r_ ) is, the more conservative is therespective strategy. Interpreting Fig. 6 leads to the followingresult (see Appendix A.10 for a sensitivity analysis).

Result 2. For parameter values which characterize typicalsemi-arid rangelands (i.e. wG, wR, μr are small and c, σr arelarge) the long-term ecological and economic impact of a strategy(α*( r_ ), r_ ) is as follows:

(i) The more conservative the strategy, the higher the meanreserve biomass μR(t) in the future:

dlRðtÞdr

< 0 anddlRðtÞda

> 0 for all t > 1:

(ii) For high rain thresholds r_ > r_̂ , the following holds: Themore conservative the strategy, the higher the mean income μy(t)in the long-term future for t>t̂:

dlyðtÞdr

< 0 anddlyðtÞda

> 0 for all t > ̂t and r > ̂r; a < a4ð ̂rÞ:

Result 2 states that the slope of the curves in Fig. 6 ispositive throughout, as far as reserve biomass is concerned;and is positive for small α⁎( r_ ), i.e. for α⁎( r_ )<α⁎( r̂), and t> t̂, asfar as income is concerned. The higher the fraction α⁎( r_ ) ofpaddocks rested, i.e. the more conservative the strategy, the

11 Numerical details are given in Müller et al. (in press).

higher is the mean reserve biomass, if the same strategy isapplied over the whole period. This effect is in line withintuition: the more conservative the strategy, the better is thestate of the rangeland in the future. As far as income isconcerned, the argument is less straightforward. In particular,the mean income in the first period is increasing in r_ , i.e.decreasing in α⁎( r_ ) (Lemma 2). A less conservative strategyyields a higher mean income in this period, since morelivestock is kept on the rangeland. This holds for severalperiods in the near future (cf. the line for t=10 in Fig. 6b).However, in the long run (for t> t̂ ≈40), the strong grazingpressure on the pasture leads to reduced reserve biomassgrowth and less forage production in the long-term future,compared to a more conservative strategy. As a result, meanincome is smaller. This can be seen in Fig. 6b: the curves areupward-sloping for sufficiently high t≥ t̂ and sufficiently smallα⁎( r_ ). As can be seen in the figure, this effect becomes strongerin the long-term future: the curves are steeper for higher t.

Result 2 holds if the growth rates of the green and reservebiomass are low, the impact of grazing on the growth of thereserve biomass is high, and rainfall is low and highly variable.This coincides with the range of parameter values which isadequate for semi-arid rangelands, because these are fragileecosystems which are highly susceptible to degradation ifgrazing pressure is high. For very robust ecosystems or verylow stochasticity of rainfall, however, the result is not valid.

For a large fraction of resting, i.e. α⁎( r_ )>α⁎( r̂ ), a moreconservative strategy (i.e. a larger α⁎( r_ )) leads to a lower meanincome, not only in the first period (Lemma 2), but also in thefuture. In this domain of strategies, resting is already so highthat the future gains in reserve biomass from additionalresting do not outweigh the losses from lower stocking.

While Result 2 describes the dynamic long-term impact ofdifferent grazing management strategies, the followinglemma analytically extends this result by specifying thesteady-state mean values of reserve biomass and income.The steady-statemean value of reserve biomass is determinedas the fixed point of themean vegetation dynamics (accordingto Eqs. (1) and (2)). The steady-state mean value of reservebiomass, in turn, determines the steady-state mean value ofincome.12

Lemma 5. 1. For an efficient strategy (α⁎(r_ ), r_ ) the steady-statemean value of reserve biomass is

lststR ¼ max KwGwRðlR−clyð1Þða�ðrÞ; rÞÞ−dwGwRðlR−clyð1Þða�ðrÞ; rÞÞ þ d

; 0

( ); ð21Þ

and the steady-state mean value of income is

lststy ¼ lststRR

lyð1Þða⁎ðrÞ; rÞ; ð22Þ

where μy(1)(α⁎(r_ ), r_ ) is given by Eq. (8), and R is the initial value of

reserve biomass.

dynamics. The latter could lead, by chance, to irreversibleextinction of the reserve biomass in the long-run even when avery conservative strategy is applied.

Fig. 6 Relation between the grazing management strategy (given by the efficient fraction of resting α* ( r¯)) and

future mean reserve biomass μR(t>0) (in units of initial reserve biomass), as well as future mean income μy(t>0) for differentstrategies on the income possibility frontier. Parameter values are μr=1.2, σr=0.7, I ·K=8000, d=0.15, wG=1.2, wR=0.2, c=0.5,I ·R=2400.

13 Underuncertainty,positive steady-statemeanvaluesdonotmeanthat a sustainable strategywill actuallyyieldpositivevaluesof reservebiomass and income. For, by chance, a sequence of rain events mayoccurwhichdrives the reserve biomass to extinction. See Footnote 11.14 As an alternative, one could set minimum requirements atstrictly positive values, representinge.g. the levels of ‘critical naturalcapital’ and ‘subsistence income’. We have chosen zero for the sakeof analytical clarity.


2. μRstst is monotonically decreasing in r_ ,

dlststRdr

< 0; ð23Þ

while μystst assumes a maximum value at r̂ >0, such that

dlststy

dr< 0 for r > ̂r: ð24Þ


For r_ > r̂ , we thus have established the following result: Themore conservative the strategy, i.e. the lower r_ and the higherα⁎( r_ ), the higher the steady-state mean reserve biomass andincome in the long run.

As the final step in our analysis, we now relate this insightto the issue of sustainability of grazing management strate-gies. For the sake of this analysis, we understand sustainabilityin the following way.

Definition 4. A grazing management strategy (α, r_ ) iscalled sustainable, if and only if it leads to strictly positivesteady-state mean values of both reserve biomass and in-come, μRstst>0 and μystst>0.

The notion of sustainability, while expressing an idea whichseems obvious and clear at first glance, is notoriously difficult todefine in an operational way. As a result, there are amultitude ofdifferent definitions of ‘sustainability’, which reveal differentaspects and, at bottom, fundamentally different understandingsof the term(seee.g.Klauer, 1999;Neumayer, 2003; Pezzey, 1992 foradetaileddiscussion). In the frameworkofourmodel,Definition4captures essential aspects of what has been called ‘strong sus-tainability’ (Pearce et al., 1990; Neumayer, 2003). It comprises anecological as well as an economic dimension, withmean reservebiomass as an ecological indicator and mean income as aneconomic indicator. It expresses the aspect of long-term conser-vation of an ecological–economic system in the sense that thesteady-statemeanvaluesofbothreservebiomassand incomeare

strictly positive.13 In contrast, an unsustainable strategy is onethat leads to the collapse of the ecological–economic system, inthe sense that the steady-state mean value of either reservebiomass or income (or both) is zero. Definition 4 constitutes aratherweak criterion of strong sustainability, by setting themini-mumrequirementswith respect to the steady-statemean valuesof both reserve biomass and income at zero.14 Yet, it enables aclear and unambiguous distinction between sustainable andunsustainable strategies in the following manner.

Lemma 6. If c>1−d/(wGwR μr), a strategy (α⁎( r_′), r_′) exists, suchthat all efficient strategies which are less conservative ( i.e. r_ >r_′ andα⁎( r_ )<α⁎( r_′))are unsustainable and all efficient strategieswith r_ >0that are more conservative (i.e. r_ <r_′ and α⁎( r_ )>α⁎( r_′)) aresustainable.


If the impact of grazing on reserve biomass growth is verysmall, i.e. if c<1−d/(wGwRμr), all strategies are sustainable. Long-termdegradation of the pasture is only a problemat all when theimpact of grazing on the vegetation is high. In this case, there is aclear and unambiguous threshold between strategies that areconservative enough to be sustainable and strategies which arenot. FromResult 1,weknow that themore risk averse a farmer is,themore conservative is his optimalmyopic strategy. Combiningthis result with Lemma 6, we can now make a statement aboutthe relation between a risk averse farmer's myopic decision andits long-term implications in terms of sustainability.

Result 3. If the uncertainty of rainfall, σr, is large and the impactof grazing c is not too large, a sufficiently risk averse myopicfarmer will choose a sustainable grazing management strategy.



Result 3 sheds new light on the question ‘How can oneexplain that people do behave in a sustainable way?’ For,Result 3 suggests the following potential explanation. That afarmer A manages an ecosystem in a sustainable manner,while another farmer B does not, may be explained simply bya higher risk aversion of farmer A. In particular, it is notnecessary to assume that farmer A has any kind of strongerpreferences for future income or sustainability than farmer B.This result holds if (i) uncertainty is large and (ii) the impact ofgrazing is not too large. If uncertainty was small, it would onlyplay a minor role in the decision making of the farmer. Hence,even a large risk aversionwould not induce amyopic farmer tochoose a conservative strategy. If, on the other hand, theimpact of grazingwas very high, the optimal strategy of even avery risk averse myopic farmer would not be conservativeenough to ensure sustainability.

For large standard deviations of rainfall and not too largegrazing impacts, the model predicts a critical degree ρ′ of riskaversion which separates the myopic farmers choosing asustainable strategy from those choosing an unsustainableone. This critical degree of risk-aversion characterizes pre-cisely that myopic farmer who chooses the strategy (α⁎(r_ ′), r_ ′),which separates sustainable from unsustainable strategies(Result 1(ii) and Lemma 6). For the parameter values used inour numerical simulations (see the caption of Fig. 6), thiscritical degree of risk aversion is ρ′=1.85, which is well withinthe range of degrees of risk aversion commonly considered asreasonable (i.e. ρ≤4; see e.g. Gollier, 2001).15

5. Conclusions and discussion

We have developed an integrated dynamic and stochasticecological–economic model of grazing management in semi-arid rangelands. Within this, we have analyzed the choice ofgrazing management strategies of a risk averse farmer, andthe long-term ecological and economic impact of differentstrategies. We have shown that a myopic farmer who issufficiently risk averse will choose a sustainable strategy,although he does not take into account long-term ecologicaland economic benefits of conservative strategies. Theintuition behind this result is that a conservative strategyprovides natural insurance for a risk averse farmer. In yearswith good rainfall the farmer does not fully exploit thecarrying capacity of the farm. Due to the buffering function ofthe reserve biomass of vegetation he thereby can shiftincome to the next year with possibly worse conditions.The more risk averse the farmer is, the higher is the benefitfrom this insurance function and the more conservative ishis optimal strategy. A sufficiently risk averse farmerchooses a strategy which is conservative enough to besustainable.

However, one should not conclude from our analysis thatrisk aversion is sufficient to ensure a sustainable develop-

15 If the standard deviation of rainfall is small, or if the grazingimpact is very large, the threshold value of risk-aversion exceedsthis range of resaonable degrees of risk-aversion.

ment in semi-arid areas. This issue requires a variety offurther considerations. First, one could adopt a moredemanding sustainability criterion than we have used (cf.Definition 4). Second, we have focused on environmental riskresulting from the uncertainty of rainfall. Other forms of risk,e.g. uncertainty concerning property-rights, or the stability ofsocial and economic relations in general, might generate atendency in the opposite direction, and promote a lessconservative and less sustainable management of theecosystem (e.g. Bohn and Deacon, 2000). Hence, in the faceof different uncertainties, the net effect is not clear and hasto be analyzed in detail. Third, additional sources of income(say from tourism) or the availability of financial services(such as savings, credits, or commercial insurance), consti-tute possibilities for hedging income risk. For farmers, allthese are substitutes for obtaining natural insurance byconservative ecosystem management and, thus, may inducefarmers to choose less conservative and less sustainablegrazing management strategies (Quaas and Baumgärtner,2006). This becomes relevant as farmers in semi-arid regionsare more and more embedded in world trade and have betteraccess to global commodity and financial markets. Ouranalysis addressed the context of grazing management insemi-arid rangelands. This system is characterized by astrong interrelation between ecology and economic use,which drives the results. While this is a specific ecological–economic system, the underlying principles and mechanismsof ecosystem functioning and economic management arefairly general. Hence, we believe that there are similar typesof ecosystems managed for the services they provide, e.g.fisheries or other agro-ecosystems, to which our resultsshould essentially carry over.

Acknowledgments

We thank Hans-Peter Weikard, Ralph Winkler, ChristianWissel, participants of the 8th Biennial Conference of theInternational Society for Ecological Economics (Montreal, June2004) as well as the ESF-Workshop Ecological–economic modellingfor biodiversity conservation (Leipzig, September 2004), and threeanonymous reviewers for their helpful discussion and sugges-tions. Maximilian Mihm has helped improve language andstyle. Financial support from the Volkswagen Foundationunder grant II/79 628 and from the German AcademicExchange Service (DAAD) is gratefully acknowledged.

Appendix A

A.1. Mean and standard deviation of the first year's income

The rainfall r is log-normally distributed, i.e. the probabilitydensity function is

f ðrÞ ¼ 1rffiffiffiffiffiffiffiffiffiffi2ks2r

p exp −ðlnr−mrÞ2

2s2r

!: ðA:25Þ

The two parameters mr and sr can be expressed in terms ofthe mean μr and standard deviation σr, mr ¼ lnlr−

12lnð1þ

r2r =l2r Þ and sr2=ln (1+σr

2 /μr2).

Fig. 7 –Parameter values are as in Fig. 6, except for c=0.9.


Using the probability density function (A.25) of rainfall andEq. (7) for the farmer's first year income, the expected valueand the variance of the first year's income are easilycalculated. The expected value is

lyð1Þða; rÞ ¼Z l

0yð1Þf ðrÞdr ¼

Z r

0rf ðrÞdrþ ð1−aÞ

Z 1

rrf ðrÞdr

¼ lr−aZ 1

rrf ðrÞdr:

The variance is

r2yð1Þða; rÞ ¼Z ∞

0ðyð1Þ−l yð1ÞÞ2 f ðrÞdr

¼ −l2y þZ r

0r2f ðrÞdrþ ð1−aÞ2

Z 1

rr2f ðrÞdr

¼ r2r þ 2alrZ 1

rrf ðrÞdr−a2

Z 1

rrf ðrÞdr

" #2−að2−aÞ

Z 1

rr2f ðrÞdr:

A.2. Expected utility function

With the specification (Eq. (13)) of the farmer's Bernoulli utilityfunction u(y), and the assumption that income is log-normallydistributed we get (using the notationmy ¼ lnly−

12lnð1þ r2y=l

2yÞ

and sy2=ln (1+σy2 /μy2)):

Eu yÞð ¼Z ∞

0

y1−q−11−q

1

yffiffiffiffiffiffiffiffiffiffi2ks2y

q exp −ðlny−myÞ2

2s2y

!dy

¼z¼lny 11−q

1ffiffiffiffiffiffiffiffiffiffi2ks2y

q Z l

−lexpðð1−qÞzÞexp −

ðz−myÞ22s2y

!dz−1

264

375

¼exp ð1−qÞ my þ 1−q

2 s2y

−1

1−q¼ l1−qy ð1þ r2y=l

2yÞ−qð1−qÞ=2−1

1−q:

A.3. Properties of the indifference curves

Each indifference curve intersects the μy-axis at σy=0. Thepoint of intersection, μ0, is the certainty equivalent of alllotteries on that indifference curve. Hence, the indifferencecurve is the set of all ðly; ryÞa IRþ� IRþ for which

lyð1þ r2y=l2yÞ−q=2 ¼ l0: ðA:26Þ

The slope of the indifference curve is obtained by differen-tiating Eq. (A.26) with respect to σy (considering μy as a functionof σy) and rearranging:

dlydry

¼ qrylyð1þ qÞr2y þ l2y

> 0: ðA:27Þ

The curvature is obtained by differentiating this equa-tion with respect to σy, inserting dμy /dσy again andrearranging

d2lydr2y

¼ ddry

dlydry

¼ qlyðl2y−ð1þ qÞr2yÞðr2y þ l2yÞðð1þ qÞr2y þ l2yÞ3

; ðA:28Þ

which is positive, if and only if μy2>(1+ρ) σy2. Furthermore,

the slope of the indifference curves increases with risingrisk aversion,

ddq

dlydry

¼ rylyðr2y þ l2yÞðð1þ qÞr2y þ l2yÞ2

> 0:

A.4. Effective decision problem

Using a monotonic transformation of U and employingPlt¼2ð1þ dÞ1−t ¼ 1=d, the decision problem becomes

maxða;rÞ

lyð1Þð1þ r2yð1Þ=l2yð1ÞÞ−q=2 þ

1dlyðtÞð1þ r2yðtÞ=l

2yðtÞÞ−q=2: ðA:29Þ

Since rainfall is independent and identically distributed ineach year, we find from Eq. (14) that mean income is

lyðtÞ ¼ lyð1Þ 1þ awRwG 1−RK

� �Z 1

rrf ðrÞdr

" #: ðA:30Þ

When calculating the variance, we neglect terms of secondorder in wR, since this is a very small number (see also thespecification of parameters in the caption of Fig. 6). With thissimplification the variance is

ryðtÞ ¼ ryð1Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 2awRwG 1−

RK

� �Z 1

rrf ðrÞdr:

sðA:31Þ

Fig. 8 –Parameter values are as in Fig. 6, except for wG=4.


Plugging this into the decision problem (Eq. (A.29)) andagain dropping terms of second order in the growth rate of thereserve biomass, we find

lyð1Þð1þ r2yð1Þl2yð1ÞÞ−q=2 þ

1dlyðtÞð1þ r2yðtÞ=l

2yðtÞÞ−q=2

¼ lyð1Þð1þ r2yð1Þ=l2yð1ÞÞ−q=2 1þ 1

d1þ awRwG 1−

RK

� �Z 1

rrf ðrÞdr

" #" #

¼ 1þ dd

lyð1Þð1þ r2yð1Þ=l2yð1ÞÞ−q=2 1þ a

wRwG

1þ d1−

RK

� �Z 1

rrf ðrÞdr

" #

ðA:32ÞUsing the abbreviation ω=wRwG (1−R /K) / (1+δ), a mono-

tonic transformation of the objective function, i.e. multipli-cation by δ / (1+δ), and, once more, the approximation ofdropping second-order terms in ω, one obtains the proposedresult.

A.5. Proof of Lemma 1

To find the efficient strategies,we first determine the strategieswhich minimize the standard deviation of income given themean income. Out of these strategies those are efficient whichmaximize the mean income for a given standard deviation.Each point on the income possibility frontier is generated byexactly one efficient strategy, since the solution of thecorresponding minimization problem is unique.

Equivalent to minimizing the standard deviation, weminimize the variance for a given mean income,

mina;r

r2y s:t: lyz l̄y; � 2 ½0; 1�; r 2 ½0;1Þ: ðA:33Þ

For a more convenient notation, we use the abbreviations

R1ðrÞ ¼Z 1

rrf ðrÞdr and R2ðrÞ ¼

Z 1

rr2f ðrÞdr: ðA:34Þ

The Lagrangian for theminimization problem (Eq. (A.33)) is

L ¼ r2yða; rÞ þ k½lyða; rÞ− l̄y ̂�¼ ½r2r þ 2alrR1ðrÞ−a2R2

1ðrÞ−að2−aÞR2ðrÞ�d ½1þ 2axR1ðrÞ�þk½½lr−aR1ðrÞ�d ½1þ axR1ðrÞ�− l̄y ̂�:The first order condition with respect to r_ is

ar f ðrÞ½−2ðlr−aR1ðrÞÞ þ ð2−aÞr�d ½1þ 2axR1ðrÞ�−½r2r þ 2alrR1ðrÞ−a2R2

1ðrÞ−að2−aÞR2ðrÞ�d 2xar f ðrÞ ¼ −kar f ðrÞd ½1þ axR1ðrÞ�þ k½lr−aR1ðrÞ�xar f ðrÞ: ðA:35Þ

The first order condition with respect to α is

½2R1ðrÞðlr−aR1ðrÞÞ−2ð1−aÞR2ðrÞ�d ½1þ 2axR2ðrÞ�þ ½r2r þ 2alrR1ðrÞ−a2R2

1ðrÞ−að2−aÞR2ðrÞ�d 2xR1ðrÞ¼ kR1ðrÞd ½1þ axR2ðrÞ�−k½lr−aR1ðrÞ�xR1ðrÞ: ðA:36Þ

Canceling the common terms α r_ f ( r_ ) in Eq. (A.35), andplugging the result into (Eq. (A.36)) leads, with some rearran-ging, to

R1ðrÞð2−aÞr ¼ 2ð1−aÞR2ðrÞfa⁎ðrÞ ¼ R2ðrÞ−rR1ðrÞR2ðrÞ− 1

2 rR1ðrÞ:

Re-inserting (Eq. (A.34)) leads to Eq. (19), which is theunique solution of the first order conditions. σy(α⁎( r_ ), r_ ) is theminimum, since σy (α,r_ ) is maximum at the corners α=1 (withρ_ >0), or ρ_ =0 (with α<1), as can be verified easily.

Eq. (19) determines the set of strategies, which generate theminimum standard deviation for any given mean income. Thisset may include different strategies which lead to the samestandard deviation, but with different mean incomes. In such acase, we drop the strategy associated with the lower meanincome. The set of efficient strategies is determinedby (α⁎( r_ ), r_ ),where r_ is chosen from the appropriate subset Ω⊆ [0, ∞) offeasible rain thresholds.

Turning to the properties of α⁎( r_ ), for r_ =0 the numerator anddenominator of Eq. (19) are equal, hence α⁎(0)=1. For r_→∞, wehave, using L'Hospital's rule repeatedly, lim

rYla⁎ðrÞ ¼ 0. Numerical

computations for a wide range of parameters (μr, σr) resulted inqualitatively the same curves α⁎( r_ ) as shown in Fig. 3.


Plugging Eq. (19) and Eq. (A.34) into Eq. (17) and differentiatingwith respect to r_ yields:

dlyða⁎ðrÞ; rÞdr

¼ �da�ðrÞr

R1ðrÞ þ a⁎ðrÞr f ðrÞ� �

d ½1þ 2axR1−xlr�

¼ 2R21ðrÞR2ðrÞ

ð2R2ðrÞ−rR1ðrÞÞ2þ a⁎2ðrÞr f ðrÞ

" #

d ½1þ 2axR1−xlr� > 0; ðA:37Þ

since, by assumption, ω μr<1.For r

¯→0, we have lim

rPY 0

R1ð rPÞ ¼ lr; limrPY 0

R2ð rPÞ ¼ r22 þ l2r , andα*(0)=1. Inserting into Eqs. (17) and (18) yields lim

rPY 0

lyða⁎ð rPÞ; rPÞ ¼0 and lim

rPY0ryða⁎ð rPÞ; rPÞ ¼ 0.

Fig. 9 –Parameter values are as in Fig. 6, except for σr=0.05.


For r¯→∞, we have lim

rPYl

R1ð rP Þ ¼ 0 and limrPYl

R2ð rP Þ ¼ 0, and limrPYl

a⁎ð rPÞ ¼ 0. Inserting into Eqs. (17) and (18) yields lim

rPYl

lyða⁎ð r

PÞ; r

PÞ ¼ lr and lim

rPYl

ryða⁎ð rPÞ; rPÞ ¼ rr.


As shown in Appendix Eq. (A.6), limrPYl

lyða⁎ðrPÞ; rPÞ ¼ lr and limrPYl

ryða⁎ðrPÞ; rPÞ ¼ rr. This is the northeast corner of the income possi-bility frontier, sinceμy=μr is themaximumpossiblemean income(cf. Lemma 2). The slope of the income possibility frontier is

dlipfy

dry¼ dlyða⁎ðrÞ; rÞ=dr

dryða⁎ðrÞ; rÞ=dr ¼ 2ryða⁎ðrÞ; rÞdlyða⁎ðrÞ; rÞ=drdr2yða⁎ðrÞ; rÞ=dr

:

From Appendix A.5 we derive

dr2ydr

¼ −kdlydr

ðA:38Þ

where λ is the costate-variable of the optimization problem(Eq. (A.33)), which is determined by Eqs. (A.35) and (A.36),

−k ¼−2lyða⁎ðrÞ; rÞ þ ð2−a⁎ðrÞÞrð1þ 2a⁎ðrÞxR1ðrÞÞ− r2yða⁎ðrÞ;rÞ2x

1þ2a⁎ðrÞxR1ðrÞ1þ 2a⁎ðrÞxR1ðrÞ−xlr

:

Thus, we have

dlipfy

dry¼ 2ryða⁎ðrÞ; rÞ

−k: ðA:39Þ

In particular for r_ →∞, it is limrYl

−kð Þ ¼ �2lrþ2r��r2r 2x1�xlr

¼ l. Hence,

limrPYl

dlipfy

dry¼ 0:

For r_ →0 both the mean income μy(α⁎( r_ , r_ )) and thestandard deviation of income σy(α⁎( r_ , r_ ) vanish (cf. AppendixA.6). Since both cannot be negative, this is the southwestcorner of the income possibility frontier. At this point, theslope of the income possibility frontier is

limrY0

dlipfy

dry¼ lim

rY0

lyða⁎ðr; rÞÞryða⁎ðrÞ; rÞ ¼ lim

rY0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1−a⁎ðrÞÞ2l2r ð1þ a⁎ðrÞxlrÞ2r2r ð1−a⁎ðrÞÞ2ð1þ 2a⁎ðrÞxlrÞ

s¼ lr

rr;

neglecting terms of second order in ω. For r_ =0, and any givenα, we have

lyða; 0Þ ¼ ½lr−aR1ð0Þ�½1þ axR1ð0Þ� ¼ ð1−aÞlrð1þ axlrÞ

r2yða; 0Þ ¼ ½r2r þ 2alrR1ð0Þ−a2R21ð0Þ−að2−aÞR2ð0Þ�½1þ 2axR1ð0Þ�

¼ ð1−aÞ2r2r ð1þ 2axlrÞ;

i.e., for small ω, the straight line between (μy, σy)= (0, 0) (α=1)and (μy, σy)= (μr, σr) (α=0) is always within the incomepossibility set. Since for r_=0 the standard deviation ismaximum for given mean income (cf. Appendix A.5), theincome possibility frontier is located above this straight line.

We have numerically determined the income possibilityfrontier for a large variety of parameters μr, σr, and ω. Theresults have provided strong evidence that under any set ofparameters the income possibility frontier is divided into twodomains: a convex domain for small σy and a concavedomain for large σy. For very small σr, these two domainsmaybe separated by a jump in the income possibility frontier,such that, in these extreme cases, the left borders of therespective income possibility sets inwardly curved to theright.


To prove part (i), we show that (a) the optimal indifferencecurve is convex over the whole range σy∈ [0, σr], and (b) theoptimum is within the concave domain of the incomepossibility frontier.

Ad (a). Rearranging Eq. (A.26) yields the following expres-sion for the optimal indifference curve (where μ0⁎ is thecertainty equivalent for the optimum)

ryly

!2

¼ lyμ0⁎

� �2=q

−1: ðA:40Þ

Inserting in the condition for the convexity of theindifference curve yields

lyry

� �2

> 1þ q⇔lyμ0⁎

<2þ q1þ q

� �2=q

: ðA:41Þ

By assumption, this condition is fulfilled for μy=μr on theindifference curve which intersects (μr, σr), i.e. which is belowthe optimal one. Since μy≤μr for all efficient strategies, thiscondition is fulfilled for all μy on the optimal indifferencecurve.

Ad (b). The minimum slope of the income possibilityfrontier in the convex domain (i.e. at the southwest border)is μr /σr (Lemma 3). The slope of the indifference curve at theoptimum (μy⁎, σy⁎), however, is smaller,

1þ q <lrrr

2<

lrrr

μy⁎σy⁎

Zq

1þ ð1þ qÞ σy⁎2

μy⁎2

<lrrr

μy⁎σy⁎

⇔qσy⁎μy⁎

μy⁎2 þ ð1þ qÞσy⁎2<

lrrr

;

¯

Fig. 10 –The threshold value of risk aversion, abovewhich amyopic farmer chooses a sustainable strategy. On the left hand sideplotted against the standard deviation σr of rainfall, on the right hand side plotted against the impact of grazing on vegetation.For cV c� ¼ wGwRlr−d

wGwRlr, all strategies are sustainable (Lemma 6). The remaining parameter values are as in Fig. 6.

16 For the two parameters K and R, no substantial influence is tobe expected: they just rescale the problem.17 This method, by stratifying the parameter space into N strata,ensures that each parameter has all proportions of its distributionrepresented in the sample parameter sets.18 To illustrate them, additional calculations were done, whereone parameter was chosen differently from the original para-meter set of Fig. 6 in each case.


where the inequality μr /σr<μy⁎ /σy⁎ holds as a consequenceof Lemma 3, and the expression on the left hand side ofthe last inequality is the slope of the indifference curve atthe optimum (cf. Eq. (A.27)). Hence, the optimum cannotbe in the convex domain of the income possibilityfrontier.

Ad (ii). For ρ=0, the indifference curves are horizontallines. Hence, the maximum of the income possibilityfrontier, which is at the corner (μy,σy)= (μr, σr), is theoptimum.

For ρ>0 corner solutions are excluded. At the corner ( μy,σy)= ( μr, σr) the slope of the income possibility frontier is zero(Lemma 3), whereas the indifference curves have a positiveslope, provided ρ>0. At the corner ( μy, σy)= (0, 0), the incomepossibility frontier is increasing with a slope μr /σr (Lemma 3),but the slope of the indifference curves is zero for σr=0 (cf.Appendix A.3).

A.9. Proof of Result 1

We have shown that the unique optimum is in the concavedomain of the income possibility frontier (Appendix A.8), andthat the slope of the farmer's indifference curves increaseswith ρ (Appendix A.3). Thus, the optimal mean income μy⁎decreases if ρ increases. Since for efficient strategies themeanμy⁎ is increasing in r

¯, the rain threshold r

¯⁎ of the optimal

strategy decreases if ρ increases.

A.10. Sensitivity analysis of Result 2

The aim of this Appendix is to show in a sensitivity analysishow the qualitative results shown in Fig. 6 and stated in Result2 depend on the parameters of the model. The sensitivityanalysis was performed using a Monte Carlo approach,repeating the computations with multiple randomly selectedparameter sets. We focused on three parameters, namely thegrowth parameter of green biomass wG, the influence c ofgrazing on the growth of reserve biomass, and the standarddeviation σr of rainfall. The other parameters either affect theoutcomes in the same direction as the selected parameters(this is the case for the growth parameter of the reservebiomass wR and the expected value of rainfall μr), or in theinverse direction (this is the case for the death rate of the

reserve biomass d).16 Hence their variation enables no furtherinsights.

A sample size ofN=20 parameter sets was created accordingto the Latin Hypercube sampling method (Saltelli et al., 2000).17

The three parameters were assumed to be independentuniformly distributed, with 0≤wG≤5, 0≤σr≤2.4 and 0≤c≤1, theupper bounds for wG and σr are guesses which proved to besuitable. The respective simulation results were compared tothe results shown in Fig. 6. The following types of long-termdynamics of mean reserve biomass and mean income (distinctfrom those stated in Result 2) were found:18

(i) If the growth parameter of the green biomasswG is verylow, i.e. if wG ·wR<d, the reserve biomass is not able topersist at all. Keeping livestock is not possible, inde-pendent of the chosen grazing management strategy.

(ii) If the impact c of grazing on the growth of the reservebiomass is very high, the mean reserve biomassdeclines to zero in finite time, unless the grazingmanagement strategy is very conservative. This isillustrated in Fig. 7, where we have chosen c=0.9.

(iii) If the growth parameter of the green biomass is veryhigh or the impact of grazing on the growth of thereserve biomass is very low, the future mean income isthe higher the less conservative the strategy is, i.e.resting is not required to preserve the ecosystem. This isillustrated in Fig. 8 for a very high growth rate of thebiomass, wG=4. Qualitatively the same outcome arisesfor very low c (see also Müller et al., in press).

(iv) If the standard deviation of rainfall σr is very small,resting is almost deterministic: for r

¯>μr, resting will

take place in hardly any year, such that mean reservebiomass μR and mean income μy are independent of thestrategy. For r<μr, restingwill take place in almost every


year, i.e. the fraction α⁎( r_ ) of rested paddocks deter-mines the outcome, as illustrated in Fig. 9 for σr=0.05.


In order to determine the steady-state mean value Rstst of thereserve biomass, we plug Eq. (1) into Eq. (2) and take theexpected value on both sides of the resulting equation. In thelong-term, the expectation value of Rt

i and Rt+1i are the same

and equal to Rstst. Given that in the long-term each camp willbe rested with equal probability, we derive

d Rstst 1þ Rstst

K

� �¼ wRwGRstst 1−

Rstst

K

� �lr−clyð1Þða⁎ðr; rÞÞ:

This equation is solved by Rstst=0 and by

Rstst ¼ KwGwRðlR−c lyð1Þða⁎ðrÞ; rÞÞ−dwGwRðlR−c lyð1Þða⁎ðrÞ; rÞÞ þ d

: ðA:42Þ

If it is positive, the last expression is the solution; otherwiseRstst=0 is the solution, since the reserve biomass cannotbecome negative. It is easily confirmed that Rstst is monoton-ically decreasing in μy(1). With a very similar argument as inLemma 2, it is shown that μy(1) is monotonically increasing inr¯. Hence, Rstst is monotonically decreasing in r

¯.

Income in each year is given by y(t)=R(t) / (I R) Σi=1I xi r. Given

that each camp is equally likely to be rested in the long-term,the long-term expected value of income is

lststy ¼ lststRR

lyð1Þða⁎ðrÞ; rÞ: ðA:43Þ

The unique interior extremum for which Rstst>0 is given by

l̂yð1Þ ¼wGwRlr þ d−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 dðwGwRlr þ dÞp

c wGwR: ðA:44Þ

Is a maximum, since for both corners μy(1)=0 and μy(1)=μrwe have μystst=0. Since μy(1) is monotonically increasing in r_ , aunique r_̂ exists, for which μy(1)= μ̂y(1).


If c>1−d / (wG wR μr), μRstst=0 for r_ →∞, by Lemma 5. That is, astrategy without resting is unsustainable. If, however, r_ →0,μy (1)=0 (by Lemma 2). Hence, as d<wG wR μr, the strategy withcomplete resting is sustainable. By Lemma 2 μy(1) is monoton-ically increasing with r

¯, which concludes the proof.

A.13. Proof of Result 3

By Lemma 6, all strategies are sustainable if cV c� ¼ wGwRlr−dwGwRlr

.Hence, even the strategy chosen by risk-neutral farmers issustainable. The interesting case is c>c̄. In that case, thestrategy chosen by a risk-neutral farmer is unsustainable.What remains to be shown is that for sufficiently large σr andsufficiently small c, a ρ′ exists, such that all farmers with riskaversion ρ>ρ′ will choose a sustainable strategy. A necessaryand sufficient condition for this statement is that

limqYl

lyða⁎ðr⁎ðqÞÞ; r⁎ðqÞÞ <wGwRlr−dc wGwR

; ðA:45Þ

where ((α⁎( r_ ⁎(ρ)), r_ ⁎(ρ)) is the optimal strategy for a myopicfarmer with risk aversion ρ. For, if Condition (Eq. (A.45)) holds,the strategy chosen by an infinitely risk averse farmer issustainable (cf. Lemma 6). Condition (A.45) is fulfilled, if (i) theright hand side is large enough and (ii) the left hand side issmall enough. The right hand side is large, if c and d are small.The right hand side is small, if σr is large compared to μr. Thishas been shown in Appendix A.7: if σr is large, the income-possibility frontier is very flat in its concave domain. Hence,the optimal μy is only slightly smaller thanμr, andCondition (Eq.(A.45)) is violated, unless c is very small. In Fig. 10, the thresholddegree of risk aversion is plotted against σr (left hand side) and c(righthandside). For both, lowσr andhigh c, this thresholdvalueexceeds plausible values of ρ. But for high σr and comparativelylow c, the threshold value ρ′ lieswell within the range of degreesof risk aversion which are commonly considered as reasonable( ρ≤4; see, e.g., Gollier, 2001).

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