MANAGEMENT SCIENCEVol. 65, No. 8, August 2019, pp. 3654–3672

http://pubsonline.informs.org/journal/mnsc/ ISSN 0025-1909 (print), ISSN 1526-5501 (online)

Socially Beneficial Rationality: The Value of Strategic Farmers,Social Entrepreneurs, and For-Profit Firms in CropPlanting DecisionsMing Hu,a Yan Liu,b Wenbin Wangc

aRotman School of Management, University of Toronto, Toronto, Ontario M5S 3E6, Canada; bCollege of Management and Economics,Tianjin University, Tianjin 300072, China; cCollege of Business, Shanghai University of Finance and Economics, Shanghai 200433, ChinaContact: ming.hu@rotman.utoronto.ca, https://orcid.org/0000-0003-0900-7631 (MH); yan.liu@tju.edu.cn,

https://orcid.org/0000-0002-6370-0683 (YL); wang.wenbin@shufe.edu.cn, https://orcid.org/0000-0003-0760-8206 (WW)

Received: October 10, 2016Revised: September 24, 2017; March 27, 2018;April 29, 2018Accepted: May 19, 2018Published Online in Articles in Advance:April 12, 2019

https://doi.org/10.1287/mnsc.2018.3133

Copyright: © 2019 INFORMS

Abstract. The price fluctuation in agricultural markets is an obstacle to poverty reductionfor small-scale farmers in developing countries. We build a microfoundation to study howfarmers with heterogeneous production costs, under price fluctuations, make crop-planting decisions over time to maximize their individual welfare. We consider bothstrategic farmers, who rationally anticipate the near-future price as a basis for makingplanting decisions, and naıve farmers, who shortsightedly react to the most recent cropprice. The latter behavior may cause recurring overproduction or underproduction, whichleads to price fluctuations. We find it important to cultivate a sufficient number of strategicfarmers because their self-interested behavior alone, made possible by sufficient marketinformation, can reduce price volatility and improve total social welfare. In the absence ofstrategic farmers, a well-designed preseason buyout contract, offered by a social entre-preneur or a for-profit firm to a fraction of contract farmers, brings benefit to farmers aswell as to the firm itself. More strikingly, the contract not only equalizes the individualwelfare in the long run among farmers of the same production cost, but it also reducesindividual welfare disparity over time among farmers with heterogeneous costs regardlessof whether they are contract farmers or not. On the other hand, a nonsocially optimalbuyout contract may reflect a social entrepreneur’s over-subsidy tendency or a for-profitfirm’s speculative incentive to mitigate but not eliminate the market price fluctuation,both preventing farmers from achieving the most welfare.

History: Accepted by Vishal Gaur, operations management.Funding: M. Hu received support from the Natural Sciences and Engineering Research Council ofCanada [Grant RGPIN-2015-06757]. Y. Liu and W. Wang received support from the NationalNatural Science Foundation of China [Grants 71702125 and 71501121, respectively].

Supplemental Material: The online appendices are available at https://doi.org/10.1287/mnsc.2018.3133.

Keywords: forward-looking farmers • contract farming • poverty reduction • socially responsible operations • cobweb phenomenon

1. IntroductionIn developing countries, small-scale farmers are amongthe poorest, and they are faced with considerableearning uncertainty.One of themain obstacles to povertyreduction for them is the price fluctuation in agriculturalmarkets. In China, for example, farmers who plant fruitor vegetables (such as watermelon, oranges, loquat,gourds, cabbage, etc.) often go bankrupt after a richharvest because prices are slashed in local markets(Feng 2014), and in India, small local onion and potatogrowers suffered a more than 50% price drop from2015 to 2016 (Buradikatti 2016, Sharma 2016). In fact,dramatic price fluctuations, far from being rare, arewidely observed in agricultural markets. Accordingto a report published jointly by a number of interna-tional organizations, such as the Food and Agricul-ture Organization, World Bank, and World Trade

Organization, agricultural markets are intrinsicallysubject to greater price variation than other markets(FAO et al. 2011). From 1983 to 1997, the market pricesfor Robusta coffee beans fluctuated by 40% to 195%of the average (Brown et al. 2008), and the prices ofwatermelon in local Chinese markets (see Figure 1)show evidence of strong cyclic annual price swings. Asthe price fluctuation is a significant obstacle to welfareimprovement of small-scale farmers, in this paper westudy strategies, operated by individuals or institu-tions, that can stabilize prices and increase small-scalefarmers’ welfare.The explanation of perpetual market price fluctua-

tions in the agricultural market has been developedsince the celebrated, macroeconomic, cobweb theorem(Ezekiel 1938). That is, the production is determinedby the farmers’ response to the price, and farmers often

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make shortsighted crop-planting decisions based onthe relatively easily obtained prices of the previousseason or blindly do what other farmers are doing (thelatter action even further amplifies that irrationality).Bariyo (2014) reported farmers’ back-and-forth de-cisions to abandon and then resume coffee cultivationon the basis of the latest market prices. Making crop-planting decisions in such a way is easy for farmersand seems logical to them, but it ignores the impacton the crop supply and the resulting market priceand, therefore, often leads to price fluctuations. Hereare two recent examples that manifest this cobwebphenomenon:

Example 1. The unprecedented high market price ofapples before 2015 induced some farmers in China toexpand their planting areas. The resulting overpro-duction led to fierce market competition and a sig-nificant price drop in 2015 (ChinaNational Radio 2015).

Example 2. The high prices of other staples have en-couraged many small farmers in Brazil to stop plantingbeans, which led to the bean shortage that has sent beanprices skyrocketing in 2016 (Parkin and Lewis 2016).

In both examples, farmers who follow the latestmarket price suffer a welfare loss later because ofthe overproduction of the crops they plant, and theymiss the chance to plant other crops that would havebeen more profitable. The extant literature on thecobweb phenomenon often takes the supply-and-demand curves as exogenously given. However,these curves are determined by farmers’ individualincentives and planting decisions. To study welfare-improving mechanisms, we start with the individualfarmer’s utility maximization problem. We first showhow farmers’ (involuntarily) shortsighted behaviorleads to a recurring boom and bust, which provides

a microfoundation for the cobweb phenomenon. Thenwe further study sustainable and incentive-compatiblemechanisms that are applied to a fraction of farmersbut can increase farmers’ total welfare.The recurring slashing of prices and loss in farmers’

welfare are attributed largely to farmers’ difficulty inobtaining market information, predicting future prices,and assessing the effect of their planting decisions onthemselves. There have been repeated efforts by gov-ernments, firms, and nongovernmental organizations(NGOs) to prevent farmers from behaving shortsight-edly. For instance, governments, NGOs, and businesssectors adopt information and communication tech-nology to help disseminate information, such as his-torical and current market prices, weather information,and advisories about near-future market prices, tofarmers (see examples in Chen and Tang 2015). Theeffectiveness of such practices is unclear. This is becausenot all farmers can obtain or process the market infor-mation, partly because of the limited access to educa-tion in developing countries (Epstein and Yuthas 2012).It is very likely that only a small fraction of farmers areable to use the market information to predict the near-future market prices and make planting decisions thatmaximize their individual welfare. Because the farmerswho can do so are self-serving, it is not clear whethertheir decisions also improve the welfare of the otherfarmers or make it even worse.We begin our analysis by building a stylized mul-

tiperiod model that captures the involuntarily irra-tional behavior of some farmers who base their plantingdecisions on the previous season’s market price.Building on this model, we study the impact of thoseself-serving, forward-looking farmers, referred to asstrategic farmers, on the welfare of the other farmerswho are shortsighted, referred to as naıve farmers. Wefind that the strategic farmers’ ability to predict the

Figure 1. (Color online) Watermelon Price Fluctuation

Source. Data from WIND Database (www.wind.com.cn).Notes. Average selling price of watermelon in the third and fourth quarter in China from 2006 to 2017. The prices are discounted by averageinflation rate.

Hu, Liu, and Wang: Socially Beneficial RationalityManagement Science, 2019, vol. 65, no. 8, pp. 3654–3672, © 2019 INFORMS 3655

near-future market price can help stabilize the marketprice, thereby benefiting farmers who have no pro-duction cost advantage. This implies that the strategicfarmers’ forward-looking behavior can be both self-serving and socially beneficial. On the other hand, weshow that the benefits are greatly discounted when thenumber of strategic farmers is small or when naıvefarmers have limited savings and would exit farmingpermanently because of bankruptcy. Therefore, it isimperative to explore other solutions.

Contract farming is another way to offer farmersa guaranteed market outlet and reduce their welfareuncertainty. There is a preseason procurement con-tract between the farmers and the buyer with a spe-cific procurement price and date for near-futuretransactions. In practice, most of the contract-farmingprojects are initiated by social entrepreneurs (SEs)and for-profit firms. For instance, Starbucks hascontract-farming agreements with local farmerswho plant coffee beans in Thailand and Indonesia(Rungfapaisarn 2013). Unlike government and NGOswho give free aid to the poor, for-profit firms haveprofit-making goals, and SEs balance social objec-tives and for-profit goals. It is reported that many SEsin the agricultural business expect the procurementcontracts to fight poverty and guarantee a reliablesupply of agricultural products (Fraser 2012). For anSE, we are interested in identifying a contract that bothimproves farmers’ welfare and brings enough profitto sustain its own operations. For a profit-driven firm,we focus on a setting in which the goal of profitmaximization is compatible with improving farmers’welfare.

Specifically, building on the microfoundation offarmers’ utility maximization, we study farming con-tracts between farmers with heterogeneous produc-tion costs and a firm (SE or profit-driven enterprise)that offers a stable procurement price to a fraction offarmers. We find that implementing a well-designedbuyout contract can help to alleviate the price fluc-tuation and bring benefits to both farmers and the SE.This stable price depends on the potential market sizeand farmers’ production costs. A lower price (i.e., in-sufficient subsidy) does not alleviate the marketfluctuation, and a higher price (i.e., over-subsidy) dis-torts the market incentive and hurts both the firmand some farmers. Moreover, we find that the optimalcontract not only equalizes the individual welfare inthe long run among farmers of the same productioncost, but it also reduces individual welfare disparityover time among farmers with heterogeneous costsregardless of whether they are contract farmers or not.This finding highlights the additional benefit of thewin–win contract in creating fairness in these regards.In making an extension of the single price contract totime-varying procurement prices, we show that the

prices in the optimal time-varying contracts convergeto the same unique price.The contract design for a for-profit firm, however,

is more subtle. In a setting in which a for-profit firmhas to pay a high price to source from an externalmarket if not satisfied by contract farmers and thelocal market, the aforementioned optimally designedcontract for the SE also increases the for-profit firm’sprofit and meanwhile improves farmers’welfare. Butit does not always maximize the firm’s profit. It islikely that the for-profit firm has a speculative in-centive to mitigate but not completely eliminate themarket price fluctuation. In such a case, the for-profitfirm alternates sourcing between contract farmersand the local market to enjoy the most cost efficiency,preventing farmers from achieving their most wel-fare. Therefore, regulation on for-profit firms may beneeded.This paper contributes to the literature that ad-

dresses the challenges of reducing agricultural marketfluctuations and improving farmers’ welfare in devel-oping economies. Specifically, we first build a micro-foundation of farmers’ utility maximization, whichenables a detailed welfare analysis. Second, with themicrofoundation, we study interactions among farmerswith heterogeneous production costs in jointly de-termining the crop’s price, which, in turn, affects theirwelfare. We show that self-interested behavior by in-dividual farmers can be socially beneficial. Third, wedesign procurement contracts for social entrepreneursor profit-driven firms that are offered to a fraction offarmers and are win–win for both the organizationand farmers. In sum, our results demonstrate that self-interested behavior by individual farmers, social en-trepreneurs, or profit-driven firms can be sociallybeneficial at the same time, improving the total socialwelfare with carefully disclosed information or opti-mally designed contracts offered to a fraction offarmers.

2. Literature ReviewOur paper is closely related to the works on the cobwebphenomenon in the macroeconomics and agronomicsliterature. Ezekiel (1938) first attributes the phenome-non (recurring cyclical herding in production) to thefact that producers often base their short-run pro-duction plans on the assumption that the present priceswill continue. Our paper differs from that literature inthree ways. First, our goal is to illustrate that the self-interested behavior of strategic farmers, social en-trepreneurs, and for-profit firms can reduce the pricefluctuations, whereas the focus of the cobweb theoryis to explain the price oscillations observed in variousmarkets. Notably, we take the perspective that suchprice oscillations can be mitigated by carefully de-signed, incentive-compatible mechanisms.

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Second, one key feature of our model is that we builda microfoundation of an individual farmer utilitymodel with heterogeneous production costs, such thatthe supply curve is derived from farmers’ utilitymaximization rather than being exogenously assumedas it is in the cobweb models. With such a utility-basedmodel, we can also evaluate each individual farmer’swelfare. For example, we show that a carefully de-signed contract offered to a fraction of farmers benefitsall stakeholders and, at the same time, reduces indi-vidual welfare disparity among farmers.

Third, the cobweb models lead to the discussionabout the rational expectations assumption and itsvalidity for all farmers (see, e.g., Nerlove 1958). Thisassumption is often made in the operations literatureon strategic consumers (see, e.g., Su 2010 and Cachonand Swinney 2011 and references therein). We considera fraction of farmers who have rational expectations orwho contract with social entrepreneurs or for-profitfirms. This feature emphasizes interactions via themarket prices among farmers who have different ra-tionality or access to external farming contracts. Ina different context but a similar spirit, Su (2010) andCachon and Swinney (2009) consider the interactionsbetween myopic and forward-looking customers, andHu et al. (2017) consider the interactions among cus-tomers who have different granularity of informationabout the service-system congestion. The latter showsthat, in the presence of information cost, a fraction ofcustomers may intentionally ignore the real-time in-formation, thereby benefiting social welfare. Aflakiet al. (2015) find that, in the presence of hassle cost,a fraction of forward-looking customers may inten-tionally choose not to search for more information, andthis rational ignorant behavior may benefit the societyas well.

We also contribute to the economics and opera-tions management literature on “social herding,” asour model captures the farmers’ shortsighted herd-ing behavior in planting or abandoning crops.Veeraraghavan and Debo (2009) identify the herdingeffect in queues in which consumers infer servicequality from the length of the queue. In our model,herding is an outcome of farmers’ shortsighted reactionsto recent prices, and we study the role of self-servingstrategic farmers and firms in alleviating herding incrop planting. We provide a solution under which theherding effect can be eliminated by offering a carefullydesigned contract, which benefits both farmers andfirms. More broadly, this paper is related to the emerg-ing literature on social operations management, whichfocuses on how social interactions leading to collectivesocial behaviors (in this paper, individual farmers’planting decisions collectively determine the total pro-duction quantity and market price) have an im-pact on firms’ operational decisions (in this paper,

procurement) and how operational decisions (in thispaper, contract farming) can influence the formation ofcollective social behaviors.Our study also links to the literature showing that

a lack of information can lead to endogenized uncertaintyin business practices. For example, the well-known“bullwhip effect” (see Lee et al. 1997) shows that thelack of information about true customer demandcauses the variance of order quantities to increase asone moves upstream along the supply chain. Hu et al.(2015) show that sales uncertainty can occur whencustomers are socially influenced by others whenmaking purchasing decisions. Cooper et al. (2006)show that incorrect beliefs about customer behaviorcause a spiral-down effect; that is, revenues system-atically decrease over time. In contrast, we focus oninvestigating win–win mechanisms that can be im-plemented by informed individuals, a social entre-preneur or a for-profit firm, to reduce the negativeeffects of the lack of market information.Poverty alleviation in developing countries has been

studied in the burgeoning operations managementliterature that focuses on the design of socially re-sponsible operations (see Sodhi and Tang 2014 fora comprehensive review). Many of those studies ex-plore the value of disclosing market information inhelping farmers improve their welfare. For instance,Chen et al. (2013) study Indian conglomerate ITC’s newbusiness model in which the firm helps farmers toobtain various information through the E-Choupalsnetwork and allows farmers to sell directly to the firm.Chen and Tang (2015) examine the value of both privateand public signals of the agricultural market. Chenet al. (2015) investigate the incentives for knowledgesharing among competing farmers. Tang et al. (2015)examine whether, under a Cournot competition,farmers should utilize market information to optimizetheir production plans when both the market potentialand the process yield are uncertain. Belloni et al. (2016)show that a monopsonistic buyer (a downstreamprincipal in general) with private demand informationcan contract with segmented farmers (upstream agents)to restore the first-best productive efficiency. Our paperdiffers from those papers in that we focus on the sce-nario in which the market information is not availableto all the farmers. We consider some farmers who canonly react shortsightedly to the market and study howthese farmers’ welfare is affected by self-serving stra-tegic farmers and firms. By modeling farmers’ dynamiccrop-planting decisions over time, we find that the self-serving individuals and firms can also be beneficialto some shortsighted farmers and improve the totalsocial welfare. In terms of methodology, we adopta multiperiod model with interactions both within andacross periods, whereas all the aforementioned papersbuild a one-period model.

Hu, Liu, and Wang: Socially Beneficial RationalityManagement Science, 2019, vol. 65, no. 8, pp. 3654–3672, © 2019 INFORMS 3657

Finally, our paper is also related to the literature onprice protection mechanisms in agricultural markets.Nicholls and Opal (2005) review price protectionimplemented through fair trade operations and studyits effectiveness. Haight (2011) suggests that a fair-tradecertification model, which guarantees a fixed priceif the farmers meet high production standards, canbe poorly implemented and only applied to limitedtypes of crops (Fair Trade USA 2012). Annan andSchlenker (2015) study the effectiveness of federalcrop insurance in the United States. They argue thatthis government program may be a disincentive forfarmers to adapt to extreme weather. Most of thesepapers are empirical studies. In our paper, we developan analytical model to study the effectiveness of apreseason procurement contract, offered by an SE ora for-profit firm, in reducing market fluctuation andimproving farmers’ welfare. We identify many poten-tial risks associated with the contract and find thata careful design is crucial to the effectiveness of sucha contract.

3. ModelWe study a multiperiod model in which farmers makeplanting decisions for a single crop in each periodover an infinite horizon. The individual farmers areinfinitesimally small scale, and the size of the farmerpopulation is normalized to one. At the beginning ofthe period t, each farmer observes pt−1, the crop’smarket price at the end of the previous period t − 1,and decides whether to cultivate the crop in thecurrent period t. The planting decision depends oneach farmer’s individual assessment of the near-futurecrop price and the cost of planting the crop. The cropproduction takes one full period. At the end of theperiod, all of the harvested crop is sold to the marketwith the price pt determined by the total output fromall farmers.

Farmers are heterogeneous in their capability ofassessing crop prices. We assume a fraction, α, of thefarmers is strategic and the remainder is naıve. Spe-cifically, the strategic farmers have full informationabout the market and can make a rational expectation ofthe near-future market price pt, but naıve farmers areincapable of correctly predicting the market price, andthey simply take pt−1 from the last period as the in-dicator of the near-future price at the end of period t.

In addition, farmers also have heterogeneous en-dowment of production costs, denoted as c, which isuniformly distributed on [0, c] capturing the idiosyn-cratic production capabilities among farmers. Let Fdenote the cumulative distribution function (c.d.f.)of production cost c. We model a farmer’s perceivedutility of planting the crop as the farmer’s individualassessed market price minus its production cost. Thatis, a strategic farmer’s perceived utility, denoted by ust ,

and a naıve farmer’s perceived utility, denoted by unt ,are

ust � pt − c, unt � pt−1− c, (1)

respectively. We focus on small-scale farmers andpremise that farmers are unlikely to collaborate tomake joint production decisions given their heteroge-neity along the two dimensions as just mentioned.In the beginning of each period, a farmer chooses tocultivate the crop if and only if the farmer’s perceivedutility is no less than zero. Our model could accom-modate the heterogeneity in farmers’ outside optionsas well by interpreting c as a general cost, includingfarmers’ opportunity cost of farming. For instance, ccould represent the heterogeneous earning farmerswould have received if they left their land and trav-eled to a city to become migrant workers. Moreover,we extend our model by considering endogenizedoutside options, the values of which are based on thefarmers’ individual perceived utility of planting othercrops (see Section 6.1).As farmers are infinitesimal and the size of the

farmer population is scaled to one, the total amountof crop produced by all strategic farmers and naıvefarmers in period t, denoted by qst and qnt , are

qst � αP(ust ≥ 0) � αF(pt),qnt � (1 − α)P(unt ≥ 0) � (1 − α)F(pt−1). (2)

As observed in many developing countries, it is noteconomical for small-scale farmers to store or transporttheir crops, especially perishable crops, to distantmarkets. Thus, in our model, we assume all the crop issold to a market that all the farmers have access to. Weuse a deterministic model to study how to influencefarmers’ shortsighted behavior, leaving aside the un-controllable yield uncertainty, which is taken into ac-count in Section 6.2. Let Ω denote the potential marketsize. As farmers sell the homogeneous crop, the marketwould be cleared at a market-clearing price associatedwith the total production quantity:

pt � Ω − bqt � Ω − b(qst + qnt )� Ω − b[αF(pt) + (1 − α)F(pt−1)], (3)

where b> 0 measures how the market price is sensitiveto the production quantity and, in particular, every unitincrease in the total supply quantity would lead themarket price to drop by $b. The larger the value of b, themore sensitive the market price to the productionquantity. Note that pt appears in both sides of thisequality, implying that the strategic farmers rationallyanticipate the future price when making planting de-cisions. Equations (1)–(3) indicate that farmers interactindirectly with one another through the market price.That is, the collection of all individual farmers’ planting

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decisions determines the market price, which, in turn,affects the realized utility of each one of them. For easeof exposition, we make the following assumptionsthroughout the paper:

Assumption1 (Diversity inPlantingDecisions). c ≤Ω≤ b + c.

Assumption 1 rules out some trivial cases: Ω≥ cavoids the situation in which some farmers will neverplant the crops regardless of the market price;Ω − b≤ celiminates that all farmers will plant crops even at thelowest possible market price.

Assumption 2 (Regular Initial Price). (a) 0≤Ω − b≤ p0 ≤Ω,and (b) p0 ≤ c.

Assumption 2 regulates the initial market price.Assumption 2(a) is a natural assumption because pt �Ω − bqt and 0≤ qt ≤ 1 for t≥ 1. We could further narrowthe range of p0 as stated in Assumption 2(b) withoutloss of generality (WLOG). This is because (1) ac-cording to Equations (2) and (3), p0 influences thefarmers’ crop production q1 and market price p1 onlythrough F(p0), and (2) for any p0 > c, we have F(p0) �F(c) � 1 because F( · ) is the c.d.f. with support on [0, c].Hence, any price beyond c is essentially equivalent to cin the sense that they have the same influence on thedynamics.

In what follows, we find the connection betweenfarmers’ naıve behavior and cyclical market fluctua-tions and examine the role of strategic farmers ininfluencing the farmers’ planting decisions and themarket prices. In Sections 4 and 5, we focus anew onsocial entrepreneurs and for-profit firms that self-interestedly induce naıve farmers not to make short-sighteddecisions.NowweuseEquations (1)–(3) to derivethe market price evolution. Define

p ≔c

c + bΩ. (4)

As we will see, without market interferences, themarket price fluctuates around p; that is, if p0 > p, thenp1 < p and vice versa. All our results in the rest wouldhold for a systemwith p0 > pwhen the potential marketinterferences start from period 2. Hence, WLOG, weassume

Assumption 3 (WLOG). p0 < p.

In fact, Assumptions 1–3 can be combined into one:max{0, c − b} ≤Ω − b≤ p0≤ p.

Proposition 1 (Market Dynamics). Suppose Assumptions1–3 hold. Denote g(α)≔ b−αb

c+αb .(i) (Convergence). If g(α)< 1, that is, α> b−c

2b , the marketprice process follows

pt � p − g(α) (pt−1 − p) � p + [−g(α)]t(p0− p). (5)

Then the market price process converges to p; that is,limt→∞ pt � p.(ii) (Divergence). If g(α) ≥ 1, that is, α≤ b−c

2b , the marketprice does not converge. In particular,

(a) If g(α) � 1, that is, α� b−c2b , the market price process

alternates between two prices p2i−1 � 2p − p0 and p2i �p0 < p2i−1 for any i ≥ 1, which are centered at p.

(b) If g(α)> 1, that is, α< b−c2b , there exists a threshold

i′ ≥ 1 such that for t< 2i′ − 1 the market price process di-verges according to Equation (5), and for t≥ 2i′ − 1, theprocess alternates between two constant prices p2i−1 � Ω −b[α + (1 − α) p2ic ] and p2i � p − g(α)(c − p)< p2i−1 for anyi≥ i′.

Proof. All proofs can be found in an online appendix.Specifically, the proof of Proposition 1 can be found inOnline Appendix E. □

Proposition 1 implies that pt − p � −g(α)(pt−1− p)alternates between positive and negative values overtime, and its absolute value | pt − p | decreases ifg(α)< 1 and (weakly) increases if g(α) ≥ 1. Thus, ifg(α)< 1, regardless of the initial price point, pt con-verges to p, which we call the limiting market price. Onthe contrary, if g(α) � 1, the price process alternatesbetween two constant prices; if g(α)>1, the process firstdiverges until a time point 2i′ − 1, after which it al-ternates between two constant prices, which corre-sponds to the scenario in which all naıve farmers floodto plant in even periods and only some of them do so inodd periods. Overall, compared with the price fluc-tuation when g(α) � 1, the price oscillates more sig-nificantly when g(α)> 1. Notably, we find that evenif we allow each farmer’s crop quantity to be a con-tinuous variable, denoted by q∈ [0, 1], as opposed todiscrete choices q∈ {0, 1} in the current model, eachfarmer would choose either q � 1 if the farmer’s indi-vidual perceived utility defined by Equation (1) is non-negative or q � 0 otherwise (see Online Appendix A).Thus, the market evolution pattern remains the same asthat described in Proposition 1.In view of Proposition 1, the size of the strategic

farmers α plays a significant role in stabilizing themarket price.

Corollary 1. Suppose Assumptions 1–3 hold. If α> 12 , the

market price process always converges. When α � 0, themarket price process converges if and only if b< c.

Note that g(α) � b−αbc+αb ≥ 0 decreases in the fraction of

strategic farmers α. Corollary 1 shows that if thereexists a sufficiently large fraction of strategic farmers(i.e., α> 1

2), the market price process always converges;that is, g(α)< 1. However, in the absence of strategicfarmers (or other forms of interference), the marketprice process converges naturally if and only if b< c,that is, when the market price is not sensitive to thetotal supply. In that situation, the market can achieve

Hu, Liu, and Wang: Socially Beneficial RationalityManagement Science, 2019, vol. 65, no. 8, pp. 3654–3672, © 2019 INFORMS 3659

self-healing, and no external help is needed to ensurelong-term stability. Therefore, we choose to omit thisrelatively simple case for expositional brevity by im-posing the following assumption throughout the restpaper. Nevertheless, most of the analysis can be gen-eralized without this assumption.

Assumption 4 (Divergence by Default). b≥ c.

More importantly, our model provides a micro-foundation of farmers’ utility maximization for theboom-and-bust cobweb phenomenon (see Ezekiel 1938):when the price realized at the end of a period is high,a large number of naıve farmersflood to plant the crop inthe upcoming period because they think the price willcontinue to be high. This results in crop overproductionand a price drop in the following period. Similarly, whenthe price realized at the end of a period is low, somenaıve farmers choose to abandon planting the crop (theymay plant another crop, see Section 6.1, or seek otheroutside options, such as going to the city as migrantworkers), thus leading to a supply shortage and a pricerebound in the following period. The naıve farmers gettrapped in the vicious cycle and suffer from the pricefluctuations caused by the recurring underproduction oroverproduction. This unfavorable situation is persistentwhen there is a lack of sufficient strategic farmers, that is,α≤ b−c

2b (equivalently, g(α) ≥ 1), and is also manifested inthe following farmers’ welfare analysis.

A farmer’s individual welfare in period t is defined asthe market price realized at the end of that periodminus the farmer’s production cost, that is, pt − c, if thefarmer plants the crop in period t and zero otherwise.Letmarket cycle i represent a pair of consecutive periods2i − 1 and 2i. We call a farmer’s average welfare in cyclei the farmer’s short-term welfare in that cycle. We firstanalyze an individual naıve farmer’s average welfarewhen the fraction of strategic farmers is relativelysmall; that is, g(α) ≥ 1. By Proposition 1, there exists i′

such that the market price process alternates between

two fixed prices for cycle i≥ i′. For brevity, our analysisof the farmer’s welfare focuses on periods after cycle i′.Denote by wn(c) and ws(c) the average welfare of anaıve farmer and a strategic farmer, respectively, withproduction cost c in any cycle i≥ i′.

Proposition 2 (FarmerWelfare in aCyclicMarket). SupposeAssumptions 1–4 hold and g(α) ≥ 1. The market price processeventually fluctuates between two constant prices p2i−1 andp2i. Consider those farmers who have a production cost c.

(i) If c≤ p2i, both the naıve farmer and the strategicfarmer plant the crop in all periods, and wn(c) � ws(c) �12(p2i−1+ p2i) − c> 0;(ii) If p2i < c≤ p2i−1, the naıve farmer plants the crop

only in period 2i and wn(c) � 12(p2i − c)< 0; the strategic

farmer plants the crop only in period 2i − 1 and ws(c) �12(p2i−1− c) ≥ 0;(iii) If p2i−1 < c, neither the naıve farmer nor the strategic

farmer plants the crop, and wn(c) � ws(c) � 0.

The microfoundation enables a detailed welfareanalysis for farmers under price fluctuations beyondthe cobweb theory. Proposition 2 shows that thefarmers’ welfare depends on their production costs.When g(α) ≥ 1, only those naıve farmers who haveextreme values of production costs would not be af-fected by the market fluctuation. Those low-cost naıvefarmers plant the crop in all periods as their cost isalways below the market price (i.e., c≤ p2i), whereasthose high-cost naıve farmers do not plant the crop inany period as their production cost always exceeds themarket price (i.e., c> p2i−1). Unfortunately, as Figure 2illustrates, the naıve farmers who have intermedi-ate cost levels (i.e., p2i < c≤ p2i−1) are hurt by the pricefluctuation because of their naıve behavior, that is,planting the crop when the latest crop price is high andabandoning the crop when the price is low. Their indi-vidual welfare is negative as shown in Proposition 2(ii).Unlike naıve farmers who have welfare loss because

they unconsciously yet effectively herd in making

Figure 2. (Color online) Farmer’s Revenue with p2i < c≤ p2i−1

Note. Naıve farmers with an intermediate cost level choose to plant the crop only in those odd periods that have realized prices below theirproduction cost, leading to negative welfare, as shown in Proposition 2(ii).

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planting decisions, strategic farmers can rationally antic-ipate thenear-futuremarket price, avoidherdingwithnaıvefarmers, and thus always receive nonnegative welfare.

The following proposition confirms that, when thereis a sufficiently large fraction of strategic farmers(i.e., g(α)< 1 or, equivalently, α> b−c

2b ), their self-servingplanting decisions always help stabilize market pricesin the long run and may improve all farmers’welfare atthe same time.

Proposition 3 (Socially Beneficial Rationality in a ConvergingMarket). Suppose Assumptions 1–4 hold and g(α)< 1.Denote by pn2i the lower price in any cycle i when all farmersare naıve farmers.

(i) In the short run (i.e., in any given cycle i), (a) eachstrategic farmer obtains a (weakly) higher surplus thana naıve farmer with the same production cost; (b) those naıvefarmers who have a production cost that is not too low, thatis, c> pn2i−2, obtain a (weakly) higher surplus with strategicfarmers than without; and (c) those naıve farmers who havea low production cost, that is, c≤ pn2i−2, obtain a highersurplus with strategic farmers than without if b � c anda (weakly) higher or lower surplus if b> c.(ii) In the long run (i.e., in the limit as the cycle i ap-

proaches ∞), (a) each strategic farmer obtains the samesurplus as the naıve farmer with the same production cost;(b) those naıve farmers who have a production cost that is nottoo low, that is, c> limi→∞pn2i, obtain a (weakly) highersurplus with strategic farmers than without; and (c) thosenaıve farmers who have a low production cost, that is,c≤ limi→∞pn2i, obtain the same surplus with strategicfarmers as without if b � c and a lower surplus with strategicfarmers than without if b> c.(iii) In the long run, if (a) b � c or (b) b> c and

p − (Ω − b) ≥ 2bc(b − c)2b2 + 3bc − 2c2

,

then the aggregate welfare of all farmers is higher withstrategic farmers than without.

Proposition 3 shows that, in both the short and longruns, the strategic farmers’ forward-looking behavior isindeed rational (parts ia and iia), and such rationalbehavior is also socially beneficial and helps increase thewelfare of naıve farmers with not too low productioncost (parts ib and iib). The welfare improvement forthose naıve farmers, who are the victims of the pricefluctuation, comes from the stabilized market priceprocess because of the presence of strategic farmers.

The naıve farmers with very low production cost,however, might obtain lower but still positive welfare inthe presence of strategic farmers when b> c (parts icand iic). This is because these farmers plant in all pe-riods, and the limiting market price p with strategicfarmers could be lower than the average price when themarket price process diverges (see Proposition 1(iib)). If

this welfare decrease is relatively small in comparisonwith the welfare gain the other farmers obtain, the totalwelfare of all farmers can be improved by the presenceof strategic farmers. We provide a sufficient conditionfor this result in part iii: the lowest possible marketprice Ω − b is sufficiently lower than the limiting pricep. This condition ensures that the population of farmerswith very low costs (which is smaller than Ω − b) isrelatively small compared with the entire population,and as a result, the total welfare increases in thepresence of strategic farmers.Although there is benefit in being forward looking,

cultivating a sufficient number of strategic farmers isstill not easy. Farmers may lack the intrinsic reasoningability or the external aids to obtain full market in-formation and may, thus, fail to make fully rationaldecisions. In Online Appendix B, we discuss the impactof backward-looking strategic behavior on the marketdynamics.In addition, when the market price process con-

verges, the fraction of strategic farmers α affects onlythe convergence rate and not the limiting market price.The larger the fraction of strategic farmers α is, thefaster the market price process converges. However,before reaching the convergence, if a farmer’s lossexceeds the farmer’s savings, the farmermaygobankruptand quit the farming business, which is considered inthe rest of this section.

Proposition 4 (Impact of Bankruptcy). Suppose Assump-tions 1–4 hold and those farmers who exit the market becauseof bankruptcy never come back. Then the market price stillconverges, but the limiting market price is (weakly) higherthan p.

Proposition 4 shows the impact of bankruptcy on thesociety: the collapse of farmers who would have sur-vived with proper interference leads to a shortage ofsupply in the long run and, therefore, a limiting marketprice that is higher than p. This situation may keepsome less cost-efficient but strategic farmers in themarket while eliminating some more cost-efficient butnaıve farmers out of the market, and may lead to a so-cially suboptimal outcome.In Online Appendix C, we detail the market dy-

namics and further present some numerical studies forthis setting with possible bankruptcy. We find that thedynamics of the market price are more complicatedwhen farmers’ exit is incorporated. There are two forcesthat make the market price converge. One is theexistence of strategic farmers who can predict the near-future price and alleviate naıve farmers’ herding be-havior. The other is farmers’ exit, which reduces thenumber of naıve farmers and, thus, reduces the impactof their irrational behavior. These two forces areintertwined with each other, and it becomes technicallychallenging to explicitly track howmany naıve farmers

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are left in the market in a given period, and thus, theprice dynamics do not have a close-form expression.We numerically show that when the fraction of stra-tegic farmers or the individual farmer’s budget is notvery small, the market price converges in a way thatfarmers who exit the market (if any) are almost or allhigh-cost farmers (whose cost is above p) and thelimiting market price is very close to or even the sameas p. In general, as stated in Proposition 4, the farmers’exit because of bankruptcy leads to a limiting marketprice that is higher than p. In view of this potentialnegative impact of bankruptcy, it is imperative to ex-amine other options that hold the potential to timelystabilize the market price and fairly distribute thebenefits of price stabilization.

4. Social EntrepreneurIn the previous section, we showed that the conversionof a sufficient fraction of naıve farmers to strategicfarmers would tilt the market dynamics from fluctu-ations to stabilization. However, one often observesthat it is very difficult to cultivate enough strategicfarmers for several reasons. First, in the case of fresh,local specialty fruits and vegetables, there is often a lackof extensive advisory systems that would help farmersobtain the market information about each crop andtrack the activities of other farmers. Second, farmers’limited ability to predict crop prices and their short-sighted behavior have remainedmajor obstacles despiteconsiderable advances in crop advisory communications.Third, external factors such as yield uncertainty may addsignificant complexity to the market-stabilizing effortsas we discuss in Section 6.2. For those reasons, in manydeveloping countries where those problems are prev-alent, it may not be as effective as it sounds to relypurely on farmers’ forward-looking behavior to sta-bilize the market. One remedy can be market stabili-zation efforts by social organizations operating in asustainable way.

In this section, we consider an SE who aims at im-proving the social welfare but also bears in mindsustaining the SE’s own operations when naıve farmersdo not voluntarily become strategic. The SE offersa fraction of farmers a preseason procurement contractthat buys out all of their crop. We refer to this contractas a buyout contract. In this contract, the SE selects andannounces a fixed buyout price po for all periods (seeOnline Appendix D.1 for time-varying buyout prices).At the beginning of each period, each farmer who isoffered the contract decides whether to sign the con-tract with the SE. Farmers who accepted the SE’scontract at the beginning of period t sell their crops tothe SE at price po at the end of period t. Right afterbuying the crops from the farmers, the SE sells them inthe market at the market price of that period, pt. The

preseason buyout contract we study here is differentfrom the minimum-price support program offered byanNGO or the government in terms of the timing of thesales commitment. In the latter, the farmers contractwith the NGO or government and can decide to selltheir crop back to them when market prices fall belowa prespecified level after the harvest, and our contractis agreed on before the season starts. The preseasoncontract can help guide the planting decisions inthe first place, whereas the minimum-price supportprogram may have a delayed effect. Although theminimum-price support program can also guaranteefarmers’ payoffs, it very likely comes at the cost ofNGOs or governments, which may not be sustainable(see, e.g., Craymer 2016). Annan and Schlenker (2015)consider moral hazard issues in minimum-price sup-port programs; that is, insurance protection reducesindividual efforts in fighting against extreme weatherconditions. We assume away such issues for the buy-out program as we focus on the impact of the priceguarantee on individual planting decisions. In thefollowing subsection, we examine whether the aidfrom the SE would bring sustainable benefits to allfarmers and to the SE as well.

4.1. Market Equilibrium in the Presence of an SEWe assume that all farmers are naıve and only anexogenous, α, fraction of farmers have access to theSE’s contract. The limited access could be attributed tomany factors. For instance, some farmers in rural areasmay be too remote to be reached, and others maydecline to work with an unfamiliar SE. We refer tofarmers who are offered the SE’s contract as type-Sfarmers and those who are not as type-N farmers.A type-S farmer’s perceived utility, if the farmer ac-cepts the contract and plants the crop, is po− c. Becausethe farmer would otherwise still behave naıvely andperceive utility pt−1− c, the farmer will accept thecontract only if the offered price po is strictly higherthan the market price pt−1 in the last period. Hence,a type-S farmer’s perceived utility in period t, denotedby uot (where the superscript o emphasizes the buyoutcontract), is

uot � max{po, pt−1} − c. (6)

In other words, farmers are perceived to be incentivecompatible in deciding whether to accept the buyoutcontract. The SE’s buyout contract aims at stimulatingfarmers to plant crops in a period when the marketprice in the last period was low. We allow the con-tract to be evoked only contingently, which providesfarmers with more flexibility than being locked in withthe SE (in Online Appendix D.2, we study a contractthat buys out a farmer’s production in all periods). The

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corresponding quantity produced by type-S farmers,denoted by qot , is

qot � αP(uot ≥ 0). (7)

The perceived utility of the remaining 1 − α fraction oftype-N farmers, unt , and their production quantity, qnt ,remain the same as in Equations (1) and (2), respec-tively. The market clearing price in period t becomes

pt(po; pt−1)≔Ω − b(qot + qnt ) � Ω − bαF(max{po, pt−1})− b(1 − α)F(pt−1).

(8)

Note that Ω − b is the lowest market price, so nofarmers would accept a contract with the buyoutprice below it. By the logic in Assumption 2(b), it iseasily shown that po makes a difference only whenit is smaller than c. Hence, we make the followingassumption.

Assumption 5. Ω − b≤ po≤ c.

Wenowstudy the long-termmarket trends for a certainlevel of buyout price po. According to Proposition 1and Corollary 1, when b≥ c and there are neitherstrategic farmers nor any contract offered by the SE,the market price oscillates over time around thelimiting market price p. The SE’s contract wouldstimulate planting the crop when the most recentmarket price is below po and, thus, reduce the pricefluctuation. The following proposition shows that theextent to which the SE’s buyout contract can help toreduce the price fluctuation varies, depending on thebuyout price po and the fraction of contract farmers α.

Proposition 5. Suppose Assumptions 1–5 hold. Denoteα0 ≔ 1− (po−p)(c+b)

(po−p0)b ,α1 ≔ 1 − cb and α2 ≔ 1− c2

b2 .(i) Suppose po> p. (a) If α>max{α1,min{α0,α2}}, the

market price process converges to p− αb(po−p)c+(1−α)b . (b) If α≤α1,

the price process does not converge.(ii) Suppose po � p. If α>α2, the price process converges

to p. Otherwise, it does not converge.(iii) Suppose po < p. The price process does not converge.

When the buyout price po is greater than the limitingmarket price p, the market price process would con-verge if α>max{α1,min{α0,α2}}, that is, there isa sufficient fraction of type-S farmers, and would di-verge if α≤α1. In fact, numerical experiments showthat there exists a threshold on α in the range[α1,max{α1,min{α0,α2}}] such that when α is greaterthan that threshold the market price process convergesto a value that is strictly less than p, and when α is lessthan that threshold the process diverges. When po isexactly equal to p, the threshold on α that determineswhether the process converges is α2. For a buyout pricepo that is below p, the SE’s contract does not stabilizethe crop price. This is because, even if there is a large

number of type-S farmers, by the time the market pricefluctuates above po no farmers would accept the buyoutcontract, and thus, the contract is no longer effectiveand the price will not continue to converge. In sucha situation, farmers are still exposed to significant pricerisks. When the buyout price is high (i.e., po> p) andprovided to a sufficient number of farmers, the SEincurs losses from the SE’s operations in the long runbecause the limitingmarket price is strictly less than thebuyout price p. Even worse, in such a situation, somenaıve farmers who are not offered the contract (i.e.,type-N farmers) may find farming no longer profitable.In short, an inadequate subsidy (i.e., low buyout price)does not eliminate the fluctuations in the market price.Too large a subsidy (i.e., a high buyout price) is notdesired, either, because it may create huge distortionsin the market, forcing many noncontract farmers togive up farming, and result in losses for the SE.The reason behind the market distortion when po is

too high is that a higher buyout price induces moretype-S farmers to produce the crop; more produc-tion leads to a lower market price and crowds out moretype-N farmers who are not protected by the buyoutcontract. This unintended consequence is exemplifiedbymany charitable programswhose generous subsidiesundermine the efforts of those unsupported poor and,thus, may end up sustaining their poverty (see Miller2014). This reveals that an SE’s efforts, which are in-spired by a desire to do good, do not necessarily result ina satisfactory outcome. A buyout price that is too highbenefits contract farmers at the expense of noncontractfarmers andmay actually create more poverty. This alsoprovides a caveat for governments and NGOs that mayhave more centralized power to offer subsidies, perhapsin the name of charity. Therefore, it is desirable to findthe optimal design of the buyout contract.

4.2. Optimal Contract DesignAs previously mentioned, the SE’s goal is to improvefarmers’ welfare while sustaining the SE’s own oper-ations. The SE’s profit in period t depends on the buyoutprice po and the market prices in periods t − 1 and t.According to Equation (6), if po > pt−1, then all type-Sfarmers with production cost c< po accept the buyoutcontract. Thus, the SE buys crops from type-S farmers inperiod t at unit price po and then sells them at unit pricept; thus, the SE’s profit is (pt − po)qot . If po≤ pt−1, then notype-S farmers accept the buyout contract in period t,and thus, the SE’s profit in period t is zero. Formally,SE’s profit in period t can be written as

πt (po; pt−1)≔ (pt − po)qot if po > pt−1,0 otherwise.

{(9)

For a given po, let wsi (c) and wn

i (c) denote the averagewelfare of, respectively, a type-S farmer and a type-N

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farmer with production cost c in cycle i. Specifically,we have

wsi (c) �

12

{1{

uo2i−1≥0}ws

2i−1 + 1{uo2i≥0

}ws2i

}and

wni (c) �

12

{1{

un2i−1≥0}wn

2i−1 + 1{un2i≥0

}wn2i

},

where

wst � pt − c if po ≤ pt−1,

po − c if po > pt−1,

{

and wnt � pt − c denote the welfare of, respectively,

a type-S and a type-N farmer in period t if the farmerplants the crop, and 1{A} is an indicator function thatequals one if conditionA holds and zero otherwise. Theindicator function is used to specify the criterion forplanting the crop; that is, the perceived utility isnonnegative. We now focus on analyzing the impact ofthe buyout contract on farmers’ welfare. See the proofof Proposition 6 in Online Appendix G for details aboutthe farmers’ welfare functions ws

i (c) and wni (c), which

are functions of pt in Equation (8) and po.The SE’s problem is to select the buyout price po that

maximizes the total welfare of farmers in the long runwhile sustaining the SE’s own operations (i.e., not in-curring a loss in any period), which can be written asfollows:

maxpo

∫ c

0limi→∞

(αws

i (c) + (1 − α)wni (c)

) 1cdc (10)

s.t. πt (po; pt−1) ≥ 0 for any t.

We assume that the SE has access to a sufficient numberof contract farmers in the remainder of this section.

Assumption 6. α>α2 � 1 − c2b2 .

If b≤ c, Assumption 6 is innocuous.Moreover,Assump-tion 6 can admit a set of feasible buyout prices thatstabilize the market (see Proposition 5). The same as-sumption is also used in Section 5 when a for-profitfirm offers a buyout contract to farmers. In the fol-lowing, we discuss the SE’s profit and farmers’welfare.Define c � p0 for b � c and c � Ω − b for b> c.

Proposition 6 (Socially Beneficial Rationality of SE).Suppose Assumptions 1–6 hold. If (1) b � c or (2) b> c andp − (Ω − b) ≥ 2bc(b−c)

2b2+3bc−2c2 , then po∗ � p is the unique optimalsolution to SE's problem (10), under which the market priceprocess converges to p and(i) The SE receives a positive profit over any finite ho-

rizon, and the profit per market cycle diminishes to zero inthe limit.(ii) Compared with the surplus that farmers would receive

in the absence of an SE,

(a) In the short run, both type-S and type-N farmers,with production cost c> c, receive a (weakly) higher surplusin the presence of an SE. The type-S farmers with productioncost c≤ c receive a lower surplus. The type-N farmers withproduction cost c≤ c receive the same amount of surplus ifb � c and a lower surplus if b> c.

(b) In the long run, both type-S and type-N farmers,with production cost c> c, receive a (weakly) higher surplusbecause of the market stabilization in the presence of an SE.Both type-S and type-N farmers with production cost c≤ creceive the same amount of surplus if b � c and a lowersurplus if b> c.

(c) In the long run, the aggregate welfare of all farmersis higher with SE than without.

Proposition 6 shows that a well-designed buyoutcontract can maximize the farmers’ total welfare whilekeeping the SE’s profit nonnegative for any period. Asshown in Proposition 6(ii), when b � c, at the individuallevel, both type-S and type-N farmers achieve certainwelfare improvement in the long run because of theSE’s contract; thus, by setting the buyout price at theoptimal as po � p, there could be a win–win–winsituation for the SE, type-S, and type-N farmers.When b> c, the welfare improvement is only at-tainable by farmers with relatively high cost c> c ,effectively balancing the individual welfare amongfarmers. The following corollaries present more de-tailed results on the virtues of this optimal buyoutcontract.

Corollary 2 (Balancing Welfare Among Farmers of theSame Cost). Suppose Assumptions 1–6 hold and the SEsets the price at po � p. Consider a farmer with productioncost c. As i approaches ∞,(a) if c≤ p, the farmer plants crops in all periods, and

limi→∞ wsi (c) � limi→∞wn

i (c) � p − c≥ 0;(b) if p< c, the farmer does not plant the crop, and

limi→∞ wsi (c) � limi→∞ wn

i (c) � 0.

In the limit, under the optimal buyout contract, themarket price converges to p (see Proposition 5(ii)),which is the same as the buyout price itself. Thus,a farmer, regardless of the farmer’s type, plants the cropin every period in the limit if the farmer’s productioncost is lower than p and otherwise does not plant inany period. Accordingly, as shown in Corollary 2, bothtype-S and type-N farmers receive identical utilities inthe long run; that is, limi→∞ws

i (c) � limi→∞wni (c).

Denote by wni (c) the average welfare of a naıve farmer

with production cost c in any cycle i in the scenariowithout an SE and with all farmers being naıve. We findthat the welfare improvement, that is, limi→∞ws

i (c) −wn

i (c) � limi→∞wni (c) − wn

i (c), however, is not monotonein the farmer’s cost endowment c. In each period, the

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optimal buyout contract po � p reduces the welfare gapbetween farmers who are at a cost advantage and whoare at a cost disadvantage.

Corollary 3 (Fairness Across Classes of Farmers withHeterogeneous Costs). Suppose Assumptions 1–6 holdand the SE sets the price at po � p. There exists i′ ≥ 1such that for any given pair of production costs c1 and c2 withc1≤ pn2i′ < c2, where pn2i′ is the market price without an SE,ws

i (c1) − wsi (c2) ≤ wn

i (c1) − wni (c2) ≤wn

i (c1) − wni (c2) for all

i≥ i′.

Farmers with low production costs below p2i′ havea cost advantage over those who have high productioncosts above p2i′ . As stated in Corollary 3, under the SE’soptimal buyout contract, the welfare gap betweenthose high- and low-cost farmers is smaller amongtype-S farmers than that among type-N farmers, whichis further smaller than that among farmers in thescenario in which all farmers are naıve. Therefore, theSE helps not only improve the total welfare but alsoreduce the welfare gap between the advantaged anddisadvantaged farmers. In sum, the distribution ofbenefits also tends to be balanced across farmers withheterogeneous production costs (see Corollary 3), inaddition to the long-run individual welfare conver-gence among farmers with the same production cost,regardless of whether they are contract farmers or not(see Corollary 2).

4.3. Contract Implementation and Discussion4.3.1. Default Risk. As shown in part iia of the proof ofProposition 6 in Online Appendix G, before the marketprice process converges, those farmers who are offeredthe contract and have a production cost less than pmayperiodically observe the realized market price higherthan the contract price po � p at which they agreed tosell to the SE. They may realize that their averagewelfare is lower than that if they plant the crop andsell it directly to the market. Thus, these farmers maychoose to default on the SE contract before the marketstabilizes. However, we find that this possible defaulthas no impact on the crop supply, and hence, themarket price process is unchanged. This is becausethese low-cost farmers will plant the crop in everyperiod regardless of whether they decide to sell thecrop in the market or to the SE. Thus, contract priceadjustment or incentives in any other form to preventthese farmers from defaulting may not be necessary.

4.3.2. Rate of Convergence. Similar to the result withstrategic farmers, under the optimal buyout contract,a larger amount of type-S farmers does not changethe market converging price but increases the marketconvergence rate and, thus, not only directly benefitsmost contract farmers themselves but also may in-directly increase the total welfare of both contract and

noncontract farmers within a short period of time.Therefore, even though approaching and convincingfarmers to participate in the contract farming can betime consuming, the SE should make the buyoutcontract accessible to as many farmers as possible if thebudget permits.However, different from the role of strategic farmers,

the SE’s contract is only effective in one period of everymarket cycle as it is executed by (naıve) farmers onlywhen the market price is low. This leads to a slowermarket price convergence rate than that under theforward-looking behavior of strategic farmers inSection 3. This finding indicates that cultivating stra-tegic farmers can be a more effective strategy over thelong run than simply offering naıve farmers a buyoutcontract.In Online Appendix D, we have also introduced two

other types of contracts that hold the potential in ex-pediting the price convergence. The first type is a time-varying contract that allows the SE to set a differentbuyout price pot for a period t. It can be shown that in thelong run the maximum of the total welfare is achievedwhen the price is stabilized at p, and there is no time-varying price contract that can further improve the totalwelfare. Hence, for the time-varying price contract, wefocus on the convergence rate instead of maximizing thewelfare as in Problem (10). In PropositionD.1we providethe time-varying contract that achieves the highest rate ofthe market price convergence. In addition, if the SE isable to endure losses, Proposition D.2 shows that themarket-limiting price can be achieved in one periodwhen α is sufficiently large. The second type is a long-term contract. If a type-S farmer accepts the contract, thefarmer commits to cultivate the crop and sell it at po to theSE in every period. We find the optimal contract is alsobeneficial to both the SE and farmers. However, we alsoidentify limitations and risks of implementing theseforms of contract in the online appendix.

4.3.3. Numerical Comparison. We numerically com-pare the market price trajectories in all the models wehave discussed so far, that is, the models with (1) allnaıve farmers, (2) α fraction of strategic farmers, (3) theoptimal fixed buyout contract, (4) the optimal time-varying contract, and (5) the optimal long-term con-tract. The latter three are offered by the SE to α fractionof type-S farmers. As shown in Figure 3, for the samegiven α, strategic farmers are the most efficient instabilizing the market, and the SE’s optimal time-varying and long-term contracts have similar perfor-mances that are better than that of the optimal fixedbuyout contract.

5. Profit-Driven EnterpriseAlthough the important role of SEs and other socialorganizations in increasing farmers’ welfare has been

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widely acknowledged, one cannot expect their partici-pation in everymarket of the agriculture industry. In thissection, we further study the commonly seen interac-tions between farmers and a profit-driven firm in agri-culture supply chains who buys raw materials fromfarmers for resale or may use these materials to makeproducts for sale. When all the farmers are naıve, wehave seen that price uncertainty is a major obstacle toa stable stream of crop supply. Given that supply un-certainty leads to operational inefficiency, reducing theprice risk and having a stablemarket supplymay benefitall the stages along the supply chain. Therefore, it is oftenin the best interest of a profit-drivenfirm to plan ahead tocontract with farmers to ensure a safe and steady supplyof agricultural products. Given that companies may notwant towait for actions taken by others, it is important toexamine how the market price risk and the resultingsupply risk can bemitigated solely by aprofit-drivenfirm.Specifically, we study the impact of a buyout contractoffered by a for-profit firm instead of an SE.

Suppose the firm has a constant demand d in eachperiod. If one unit of crop raw materials can be con-verted to one unit of final products, d can also be

interpreted as the firm’s target level of crop pro-curement. If the market supply qt ≥ d, the firm buys dunits of raw materials from the market at the marketprice pt. If qt < d, the firm has to buy an additionalquantity d − qt from an external market at a higher pricepm with pm > pt for any t≥ 0. The external market isisolated from the focal market we have studied. Theprice difference pm− pt can also be viewed as a penaltycost for those demands in excess of supply from thefocal market. We start with all farmers in the pop-ulation being naıve farmers. Just like the SE in theprevious section, the firm offers a preseason pro-curement contract at the beginning of each period witha fixed price po to α fraction of naıve farmers. As before,we refer to farmers who are offered the contract astype-S farmers and the rest as type-N farmers.

Assumption 7. α< d< Ω−pb .

Note that d< Ω−pb ensures that a stable market can

provide sufficient supply to meet the firm’s demand,and d>α states that the production by type-S farmersalone is insufficient to satisfy all of the firm’s demand.Assumption 7 ensures that the firm desires a stable

Figure 3. (Color online) Numerical Comparison of Different Scenarios Across Various α

Note. Parameters: b � 11, c � 10,Ω � 12.

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market and has the incentive to induce both typesof farmers to produce enough of the crop for thefirm.

The total procurement cost tomeet demand in periodt is

ft (po)≔po min{qot , d} + pt [min{qt, d} − qot ]++ pm(d − qt)+ if po > pt−1,pt min{qt, d} + pm (d − qt)+ otherwise,

⎧⎪⎪⎪⎨⎪⎪⎪⎩(11)

where [x]+≔max{x, 0}; pt is the market price thatevolves according to Equation (8); qot � αF(po) is theamount of crop produced by the type-S (contract)farmers; and qt is the total supply from the focalmarket,which, as an inverse function of the market price, isequal to Ω−pt

b by definition. Within period t, the firmfulfills its demand by using different sources in thefollowing pecking order. When po > pt−1 and, thus,some type-S farmers accept the contract, the firm firstpurchases at most αF(po) from contract farmers at unitprice po and then sources from the market at unit cost ptand finally resorts to the external market with unit costpm until the demand is fully satisfied. When po ≤ pt−1,the firm first buys from the market and then from theexternal market if necessary.

To increase the total quantity of potentially lesscostly supply qot � αF(po) from the farmers, the firmcould either increase α by making the contract acces-sible to more farmers or raise the buyout price po, bothof which come at a cost. (In Section 6.2, we consideryield uncertainty. When yield uncertainty is pro-nounced, the firm may want to further increase α andpo to secure supply.)

Different from the SE’s objective, the firm’s goal is tominimize its average long-term cycle procurement cost,that is,

minpo

limT→∞

1T

∑Ti�1

[ f2i−1 (po) + f2i (po)]. (12)

Farmers’ planting decisions and, hence, the marketevolution and farmers’ welfare under the buyoutcontract with a constant price are the same as specifiedin Propositions 5 and 6(ii) when the contract is offeredby the SE.

Proposition 7 (Socially Beneficial Rationality of For-ProfitFirm). Suppose Assumptions 1–7 hold. Denote by pn2i−1 thehigher price in any given cycle i when all farmers are naıvefarmers. Suppose limi→∞

Ω−pn2i−1b < d. If the buyout price is set

to po � p for all periods, then(i) The market price converges to p.

(ii) There exists an i′ > 0 such that the firm obtainsa higher profit with the contract than without in any cyclei> i′. In particular, if b � c, po � p is an optimal solution tothe firm’s problem (Problem 12).(iii) In both the short and long run, farmers’ welfare

improvement is the same as specified in Proposition 6(ii).

Proposition 7 provides sufficient conditions underwhich the buyout contract with po � p benefits theprofit-driven firm and farmers in both the short andlong terms. The market will be stabilized, eliminatingthe negative impact of farmers’ shortsighted behaviorand providing a stable and low-cost crop supply to thefirm. Specifically, limi→∞

Ω−pn2i−1b < d< Ω−p

b ensures thata stable market can provide sufficient supply to meetthe firm’s demand, whereas an unstable market maynot. As mentioned, the condition d>α in Assumption 7further ensures that the production by type-S farmersalone (at most α) is insufficient to satisfy all of the firm’sdemand. Hence, these conditions ensure that the firmhas an incentive to stabilize the market and induce bothtypes of farmers to produce enough of the crop forthe firm.Moreover, when b � c, Proposition 7(ii) shows that

the buyout contract with po � p is indeed optimalamong all buyout contracts under the long-run averageprocurement cost criterion. However, when b> c, thatis, when the market price is relatively more sensitive tothe market supply, the firm may be driven by thenature of profit-seeking to choose a different contractfrom po � p.

Remark 1 (Speculative Firm). In this remark, we illus-trate the firm’s speculative behavior and discuss whythe for-profit firm may not want to completely elimi-nate the price fluctuation. Suppose Assumptions 1–7hold and b> c. For any given market price p2i′ < p, thefirm is able to control the market prices to alternatebetween two fixed prices, that is, p2i′ and a higher pricep + b(1−α2)

c ( p − p2i′). The firm achieves this outcome bysetting the contract price to po � p2i′ + α2

α ( p − p2i′) for allt≥ 2i′. The price fluctuation enables the firm to buyfrom contract farmers at po in odd periods when themarket price is expected to be high and buy directlyfrom the market in even periods when the market priceis expected to be low.The for-profit firm gets long-term benefits from this

speculative behavior only if there is a sufficient amountof supply and the firm does not have to buy from theexpensive external market. This condition is achievedas long as the lower market price p2i′ is not too low; thatis, p2i′ ≥ p − 1

1−α2( p − cd). The firm’s profit in the long

run is greater than that when the contract price is po � p,and the increased profit comes at the expense ofa decrease of welfare for farmers with relatively highproduction costs (i.e., c≥ c).

Hu, Liu, and Wang: Socially Beneficial RationalityManagement Science, 2019, vol. 65, no. 8, pp. 3654–3672, © 2019 INFORMS 3667

6. ExtensionsThe unstable market supply resulting from pricefluctuations is often exacerbated by a host of endoge-nous and exogenous causes. For example, farmers maychoose which crop to plant among multiple optionsbased on their market prices, and this decision can beinfluenced by crop production yield, which is affectedby weather, insect pests, etc. In this section, we firstextend Section 3 to a two-crop setting in which bothnaıve and strategic farmers choose to cultivate one oftwo types of crops. In other words, the outside optionof planting one crop for a farmer is endogenized as thevalue of planting the other. We then further extend thismodel to capture the production yield uncertainty. Thesame extensions can be naturally made to the settingsin Sections 4 and 5.

6.1. Two CropsWe consider farmers who choose to plant one of twotypes of crops in each period. We useA and B to denotethe two crops. In line with the settings in the previoussections, we assume a farmer’s production costs ofplanting crops A and B are c and c − c, respectively,where c is uniformly distributed on [0, c]. In addition tocapturing the heterogeneity of farmers’ productioncosts, the assumed cost structure reflects two plausiblereasons: first, each farmer may have different produc-tion costs for different crops; second, no farmer has costadvantages over other farmers with both crops. Theanalysis can be extended to settings with productioncosts in more general forms.

We first derive the quantities of crops A and Bproduced by naıve and strategic farmers. Under theassumed production cost structure, a naıve farmer’sperceived utility of cultivating crops A and B is uA,nt �pAt−1− c and uB,nt � pBt−1− (c − c), respectively. Similarly,the expected utility of planting crop A for a strategicfarmer who rationally anticipates the market price isuA,st � pAt − c, and the expected utility of planting crop Bis uB,st � pBt − (c − c). Compared with the single cropmodel in Section 3, the value of planting one crophere can be taken as the endogenous outside optionof planting the other.1 We assume uA,τt and uB,τt forτ∈ {n, s} are all nonnegative to avoid the degenerationto the single-crop model. A naıve or strategic farmerwill cultivate crop A if uA,τt ≥ uB,τt for τ∈ {n, s} andotherwise will cultivate crop B. Thus, in period t, thequantities of crops A and B produced by naıvefarmers are

qA,nt � (1 − α)P(uA,nt ≥ uB,nt ) and

qB,nt � (1 − α)P(uA,nt < uB,nt ). (13)

Similarly, the quantities of crops A and B pro-duced by strategic farmers are qA,st � αP(uA,st ≥ uB,st )

and qB,st � αP(uA,st < uB,st ). For ease of exposition, weassume crops A and B have the same market sizeΩ.Then the market-clearing prices of crops A and Bare pAt � Ω − b(qA,nt + qA,st ) and pBt � Ω − b(qB,nt + qB,st ).Proposition 8. Suppose Assumptions 1–3 hold.(i) If g(α) ≥ 1, the market price process in each market

does not converge.(ii) If g(α)< 1, the market prices of crops A and B are

pAt � Ω − b2 − Δt(α) and pBt � Ω − b

2 + Δt(α), where Δt(α) �12(−g(α))t−1(pA1 − pB1 ), and both converge to Ω − b

2 as t→∞.

Proposition 8 shows that when farmers are able tochoose one of two crops to plant in each period, themarket evolution is similar to the patterns in the single-crop model (see Proposition 1). When there is a lack ofsufficient number of strategic farmers, the market priceprocess for each crop does not converge and eventu-ally alternates between two prices; when there isa sufficiently large number of strategic farmers, theprice process of each cropwill converge toΩ − b

2 , whichis independent of the segment size of strategic farmers,α. Comparing with the single-crop model, the keydifference is that now the farmers’ outside option ofcultivating one crop is endogenized as the value ofcultivating the other crop. In particular, the high priceof one crop at the end of a period encourages naıvefarmers to plant this crop in the next period, exacer-bating the shortage of the other crop. Hence, themarket prices for two crops tend to wax and wane inalternation.The limiting price Ω − b

2 in Proposition 8(ii) impliesthat when there are sufficient strategic farmers, even-tually half of the farmers with the cost advantage forcrop A, that is, c< c

2, always cultivate crop A, and theother farmers always cultivate crop B in each period.Farmers are divided exactly in half because the po-tential market sizes of both crops are assumed to be thesame as Ω. It can be shown that, when the two cropshave different market sizes, the market price of eachcrop will still converge as long as (1) there are enoughstrategic farmers and (2) the difference between themarket sizes of the two crops is not too large. Over thelong run, the crop with a larger market size attractsmore than half of the farmers. In terms of the farmers’welfare, the same statement as in Proposition 3 stillholds in the two-crop model because strategic farmerscontinue to help stabilize the market prices for bothcrops. That is, our finding about strategic farmers’socially beneficial rationality continues to hold for thetwo-crop model.

6.2. Random YieldNow we extend the two-crop model by consideringyield uncertainty in the farmers’ harvest. This uncer-tainty, which is attributed to uncontrollable external

Hu, Liu, and Wang: Socially Beneficial Rationality3668 Management Science, 2019, vol. 65, no. 8, pp. 3654–3672, © 2019 INFORMS

factors (e.g., weather, water supply, pests), captures thefact that the harvest quantity is often different from thequantity actually planted. We refer to the former asthe random yield and the latter as the production quantity.We capture yield uncertainty by the ratio of the ran-dom yield to the production quantity and assume thatthe yield uncertainty in period t for all crops in the samedistrict is the same, denoted by γt.

In line with the two-crop model, we denote by qtA,τ

and qtB,τ for τ∈ {n, s} the farmers’ production quan-

tity of crops A and B given the most recent marketprices pAt−1 and pBt−1, respectively. The random yieldsof the two crops are QA,τ

t � qtA,τγt and QB,τ

t � qtB,τγt

for τ∈ {n, s}.We assume that yields γt for all t≥ 1 are in-

dependent and identically distributed random var-iables with finite variance and that they are independentof farmers’ production costs. In a typical productionenvironment, γt ≤ 1, however, this is not a requirementof our model. We use γt � 1 to represent an average ornormal external condition. The case of γt > 1 can capturea growing season when crops grow very well, andγt < 1 captures a season with extreme weather orblight that dooms a harvest. We assume E(γt) � 1.

Now we derive the farmers’ production quantitiesand the corresponding market prices in a period.For naıve farmers, the production quantities (qtA,n, qtB,n)can be written in the same way as (qA,nt , qB,nt ) inEquation (13) under the assumption that E(γt) � 1. Incontrast, the production quantities of strategic farmersdepend on the rationally anticipated market prices,which are affected by the yield. Specifically, theperceived expected utilities of planting crops A and Bcan bewritten as uA,st � E [γPA

t ] − c and uB,st � E [γPBt ] −

(c − c), where γ is a representative random variablewhose distribution is the same as γt for t≥ 1 and isindependent of these γt, and PA

t and PBt are the random

market prices for period t (see below). Then the cor-responding production quantities of crops A and Bcan be expressed as qt

A,s � αP (uA,st ≥uB,st ) and qtB,s �

αP (uA,st < uB,st ). The resulting market-clearing pricesare now random, depending on the random yieldsof the two crops, that is, PA

t � Ω − b(QA,st + QA,n

t ) andPBt � Ω − b(QB,s

t + QB,nt ).

Next we show the convergence of the randommarketprice processes in a probabilistic sense.

Proposition 9. Suppose Assumptions 1–3 hold,var[ln (1−α)bγ

c+αbE[γ2]]<∞, and b(1 − 2α)E[γ2] ≤ c. Define

I(α,γ)≔ ln (1−α)bc+αbE[γ2] + E[lnγ].

(i) If I(α,γ) ≤ 0, then the random market price processesPAt and PB

t converge in probability toward the same randomvariable P

≔Ω − b

2γ; that is,

limt→∞P(∣∣PA

t −P ∣∣> ε) � lim

t→∞P(|PBt −P

| > ε)

� 0 for any ε> 0.

Moreover, the price difference between two crops convergesto zero in probability as t goes to ∞.(ii) If I(α,γ)> 0, then the random market price processes

do not converge.

As (1 − 2α)E[γ2] and I(α,γ) decrease in α> 0,Proposition 9 shows that, compared with a marketin which all farmers are naıve, the market priceprocesses are more likely to converge when there aremore strategic farmers. When the condition I(α,γ) ≤ 0holds, the price processes converge to a randomvariable P

, which is an affine transformation of the

yield uncertainty γ. If γ � 1, we recover the deter-ministic two-crop model; see Proposition 8(ii). Whenthe market prices converge to the random variableP≔Ω − b

2γ, the price difference between two cropsdiminishes, and hence, the fluctuation of market pricesremaining in the market is completely attributed tothe yield uncertainty.

7. ConclusionIn developing countries, the price fluctuation in agri-cultural markets is blocking the path to prosperity forsmall-scale farmers. Although the burgeoning litera-ture on sustainable operations studies the value ofdisclosing market information for price discovery andhow that affects farmers’ welfare, little attention hasbeen paid to farmers’ heterogeneous capability inobtaining and processing market information. In re-ality, only farmers who are educated and not in remoteareas may obtain and utilize the market information intheir decision making. It, thus, gives rise to the concernwhether efforts to disseminate market informationtreat all farmers fairly and whether the market isevolving in a socially desirable way.In this paper, we offer a stylized multiperiod model

to study farmers’ dynamic crop-planting decisions andtheir impact on the market prices. We consider bothnaıve farmers who react shortsightedly to the latestmarket price and strategic farmers who are able tocollect and utilize market information to anticipaterationally the near-future market price. Our modelbuilds a microfoundation to capture the root cause ofthe price fluctuations observed in many agriculturalmarkets. That is, because naıve farmers base theircrop-planting decisions on the latest market price, they

Hu, Liu, and Wang: Socially Beneficial RationalityManagement Science, 2019, vol. 65, no. 8, pp. 3654–3672, © 2019 INFORMS 3669

cyclically herd in planting the crop, thus leading to re-curring overproduction and price slashing. We find thatthe forward-looking behavior of strategic farmers couldalleviate the herding behavior, therefore being bothindividually rational and socially beneficial in that it canstabilize the market price process and increase farmers’total welfare. However, to sustain this notion of so-cially beneficial rationality, there must be a sufficientnumber of strategic farmers to collectively move theneedle.

Another major finding of this paper is that evenwhen it fails to cultivate strategic farmers, both socialentrepreneurs and for-profit firms can benefit farmersas well as themselves by offering a preseason buyoutcontract to a fraction of farmers. Those farmers thenonly need to base their crop-planting decisions onthe price offered by the firm. However, althougha carefully designed contract can improve the totalsocial welfare and lead to smaller individual welfaregaps among farmers, a nonsocially optimal contract

may reflect a social entrepreneur’s over-subsidytendency or a for-profit firm’s speculative incentiveto mitigate but not eliminate the market price fluc-tuation, preventing farmers from achieving the mostwelfare. Finally, we also extend our model withstrategic farmers to account for multiple crops andyield uncertainty, and our analysis confirms the role ofstrategic farmers in improving farmers’ welfare.

AcknowledgmentsThe authors thank Vishal Gaur (department editor), the as-sociate editor, and the three anonymous referees for com-ments and suggestions, and they also thank Atalay Atasu,Ying-Ju Chen, Robert Swinney, Haoying Zhang, and theparticipants at 2017 MSOM Sustainable Operations SIGconference and Early Career Sustainable Operations Man-agement Workshop for comments. The authors contributedequally to this work.

Appendix. Summary of ResultsWe summarize our main results in the following tables.

Table A.1. Price Evolution and Convergence

Naive farmers 1 − α and a fraction α of Condition Market price evolution Converges to p � Convergence rate | p2i+1−p || p2i−1−p |

(1) Naıve farmers b< c pt � p + (−bc

)t (p0 − p) p(bc

)2b≥ c No convergence

(2) Strategic farmers g(α)< 1 pt � p + [−g(α)]t (p0 − p) p g2(α)g(α) ≥ 1 No convergence

(3) Type-S farmers undercontract with price p

α>α2 p1 � Ω − α bc p − (1 − α) bc p0, p (1 − α) b2

c2

p2i−1 � p + [(1−α)b2c2

]i−1 (p1 − p),p2i � Ω − b

c p2i−1α≤α2 No convergence

(4) Type-S farmers undertime-varying contract

α>α1 p2i−1 � cc+αb

[Ω − (1−α)b

c p2i−2], p > (1 − α) b2

c2

p2i � Ω − bc p2i−1

α≤α1 NA(5) Type-S farmers under

long-term contract with price pα>α1 pt � p + [− (1−α)b

c

]t (p0 − p) p[(1 − α) bc

]2

α≤α1 No convergence(6) Backward-looking farmers

(see Online Appendix B)b< c pt � Ω − b

c

(1 − α

2

)pt−1 − b

cα2 pt−2 p NA

b≥ c NA NA

Notes. g(α)≔ (1−α)bc+αb , α1 ≔ 1 − c

b, and α2 ≔ 1 − c2b2. We omit the trivial scenarios when market prices take extremely high or low values (i.e., when all

farmers or no farmers plant the crop). NA, not applicable.

Table A.2. Farmer Welfare Summary

Scenario Short-run welfare Long-run welfare (i→∞) Comparison (c≥ c)

(1)

wni (c) �

p2i−1+ p2i2 − c if c≤ p2i−2

p2i− c2 if p2i−2 < c≤ p2i−1

0 if c> p2i−1

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ � wni (c)

(2) wsi (c) �

p2i−1+ p2i2 − c if c≤ p2i−2

p2i− c2 if p2i−2 < c≤ p2i−1

0 if c> p2i−1

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ � p − c if c≤ p0 if c> p

{> (1); > ws

i in (3);>wn

i in (2)

Hu, Liu, and Wang: Socially Beneficial Rationality3670 Management Science, 2019, vol. 65, no. 8, pp. 3654–3672, © 2019 INFORMS

Endnote1There are papers that study the farmers’ planting decisions whenthere are crop rotation benefits, such as a larger yield and lowerfarming costs on rotated farmland (see, e.g., Boyabatli et al. 2019 andrelated papers mentioned therein). Our study focuses more onfarmers’ planting decisions as reactions to themarket-price dynamics.

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Scenario Short-run welfare Long-run welfare (i→∞) Comparison (c≥ c)

wni (c) �

p2i−1+ p2i2 − c if c≤ p2i

p2i−1− c2 if p2i < c≤ p2i−1

0 if c> p2i−1

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ � p − c if c≤ p0 if c> p

{> (1); > wn

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(3) wsi (c) �

p+ p2i2 − c if c≤ p2i−2

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0 if c> p2i−1

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ � p − c if c≤ p0 i f c> p

{> (1)

wni (c) �

p2i−1+ p2i2 − c if c≤ p2i−2

p2i− c2 if p2i−2 < c≤ p2i−1

0 if c> p2i−1

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ � p − c if c≤ p0 if c> p

{> (1); > ws

i in (3)

(4) wsi (c) NA. � p − c if c≤ p

0 if c> p

{> (1)

wni (c) NA. � p − c if c≤ p

0 if c> p

{> (1)

(5)ws

i (c) � p − c if c≤ p0 if c> p

{� p − c if c≤ p

0 if c> p

{> (1); > ws

i in (3);> wn

i in (5) if c> p2i−2

wni (c) �

p2i−1+ p2i2 − c if c≤ p2i−2

p2i− c2 if p2i−2 < c≤ p2i−1

0 if c> p2i−1

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩� p − c if c≤ p

0 if c> p

{> (1); > wn

i in (3)

Notes. In scenarios (2)–(5), we assume b≥ c. For all five scenarios, we assume the market convergence conditions (in Table A.1) hold.

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