Altruistic Rationality:
The Value of Strategic Farmers, Social Entrepreneurs and
For-Profit Firms in Crop Planting Decisions
Ming Hu
Rotman School of Management, University of Toronto, Toronto, Ontario, Canada M5S 3E6, [email protected]
Yan Liu
College of Management and Economics, Tianjin University, Tianjin, China, [email protected]
Wenbin Wang
School of International Business Administration, Shanghai University of Finance and Economics, Shanghai, China 200433,
[email protected]
Price fluctuations in agricultural markets are an obstacle to poverty reduction for small-scale farmers in
developing countries. We build a micro-foundation to study how farmers of heterogeneous production costs,
under price fluctuations, make crop-planting decisions over time to maximize their incomes. We consider both
strategic farmers, who rationally anticipate the near-future prices as a basis for making planting decisions,
and naı̈ve farmers, who shortsightedly react to recent crop prices. The latter behavior may cause recurring
over- or under-production, which leads to price fluctuations. We find it important to cultivate strategic
farmers, because their self-interested behavior alone, made possible by sufficient market information, can
reduce price volatility and benefit all farmers. Surprisingly, even a tiny fraction of strategic farmers may be
enough to stabilize the market price. In the absence of strategic farmers, an optimally-designed pre-season
procurement contract, offered by a social entrepreneur or for-profit firm to a fraction of contract farmers,
brings benefit to all the farmers as well as to the firm itself. More strikingly, the optimal contract not only
brings equity in the long run among farmers of the same production cost, but also reduces income disparity
over time among farmers of heterogeneous costs, regardless of whether they are contract farmers or not. On
the other hand, a poorly designed contract may distort the market and drive non-contract farmers out of
business.
Key words : forward-looking farmers; contract farming; poverty reduction; socially responsible operations
1.
Introduction
In developing countries, small-scale farmers are among the poorest, and they are faced with considerable income uncertainty. One of the main obstacles to poverty reduction for these farmers is
the price fluctuation in agricultural markets. In China, for example, farmers who plant fruit or
1
2
vegetables (such as watermelon, oranges, loquat, gourds, cabbage, etc.) often go bankrupt after a
rich harvest because prices are slashed in local markets (China News 2014); and in India, small
local onion and potato growers suffered a more than 50% price drop from 2015 to 2016 (Tribune
India 2016, Hindu 2016). In fact, price fluctuations, far from being rare, are widely observed in
agricultural markets. According to a report published jointly by a number of international organizations (the Food and Agriculture Organization, the World Bank, the WTO, etc.), agricultural
markets are intrinsically subject to greater price variation than other markets (UNCTAD 2011).
From 1983 to 1997 the market prices for Robusta coffee beans fluctuated by 40 percent to 195
percent (International Institute for Sustainable Development 2008), and the prices of oranges in
local Chinese markets (see Figure 1) show evidence of strong cyclic annual price swings. As price
fluctuation is a significant obstacle to improvements in the welfare of small-scale farmers, in this
paper we build a stylized model to capture one driving force behind the price fluctuation observed
in agricultural markets and its impact on farmers’ welfare. Then we study strategies that can
stabilize prices and increase small-scale farmers’ welfare.
Figure 1
Orange Price fluctuation (data from WIND database www.wind.com.cn).1
4.0
RMB/kg
3.0
2.0
1.0
0.0
04/01/08
03/01/09
03/01/10
03/01/11
03/01/12
The recurring slashing of prices and loss in farmers’ welfare are attributed largely to farmers’
difficulty in obtaining market information, predicting future prices and assessing the effect of their
planting decisions on themselves. In developing countries there are few mechanisms for discovering
future market prices, especially for local specialty crops. As a result, these farmers often make
short-sighted crop-planting decisions based on the relatively easily obtained prices of the previous
season, or blindly do what other farmers are doing (the latter action even further amplifies that
1
As shown by the discontinuous data points, prices on the orange market are available only at a certain time every
year as oranges are perishable and farmers in China harvest them only once a year.
3
irrationality). Wall Street Journal (2014) reported farmers’ back-and-forth decisions to abandon
and then resume coffee cultivation on the basis of the latest market prices. Making crop planting
decisions in such a way is easy for farmers and seems logical to them, but it ignores the interaction
between crop supply and its price on the market, and therefore often leads to the fluctuations in
prices and a loss of welfare for farmers. Here are two recent examples.
Example 1. The unprecedented high market price of apples before 2015 induced some farmers in China to expand their planting areas. The resulting overproduction led to fierce market
competition and a significant price drop in 2015 (China Radio 2015).
Example 2. The high prices of other staples have encouraged many small farmers in Brazil to
stop planting beans, which led to the bean shortage that has sent bean prices skyrocketing in 2016
(Wall Street Journal 2016a).
In both examples, farmers who follow the latest market price suffer a welfare loss later due to the
overproduction of the crops they plant, and they miss the chance to plant other crops that would
have been more profitable. Our model shows how farmers’ (involuntarily) short-sighted behavior
leads to a recurring boom-and-bust in market prices and to farmers’ welfare loss. Using that model,
we study means that can help farmers increase their welfare and that, at the same time, offer
incentives to the enabling individuals or institutions.
There have been repeated efforts by governments, market institutions, and non-profit organizations (NGOs) to reduce the price fluctuations faced by farmers. For instance, to prevent farmers
from behaving short-sightedly, governments, NGOs, and business sectors use information and communication technology to disseminate information, such as historical and current market prices,
weather information and advisories about near-future market prices, to farmers (see examples in
Chen and Tang 2015). Such practices, however, may not be very effective since not all farmers can
obtain or process the market information. It is very likely that only a small fraction of farmers
are able to use the market information to predict the near-future market prices and make planting
decisions that maximize their individual welfare. Since the farmers who can do so are self-serving,
it is not clear whether their decisions also improve the welfare of the other farmers or make it even
worse.
We begin our analysis by building a stylized multi-period model that captures the involuntary
irrational behavior of farmers who base their planting decisions on the previous season’s market
price. This base model provides a micro-foundation for the celebrated, macro-economic, cobweb
phenomenon (Ezekiel 1938). Building on this base model, we study the impact of those self-serving,
forward-looking farmers (denoted as strategic farmers) on the welfare of the other farmers, who
4
are short-sighted (denoted as naı̈ve farmers). We find that, thanks to the strategic farmers’ ability
to predict the near-future market price, even a small number of strategic farmers helps to stabilize
the market price, thereby benefiting all farmers. This implies that the strategic farmers’ forwardlooking behavior is both self-serving and altruistic. However, we show that the benefits are greatly
discounted when farmers have bounded rationality. Therefore it is imperative to explore other
solutions.
Contract farming is another way to offer farmers a guaranteed market outlet and reduce their
welfare uncertainty. This is a pre-season procurement contract between the farmers and the buyer,
with a specific procurement price and date for near-future transactions. In practice most of the
contract farming projects are initiated by social entrepreneurs (SEs) and for-profit firms. For
instance, Starbucks has contract-farming agreements with local farmers who plant coffee beans in
Thailand and Indonesia (The National 2013). Unlike government and NGOs who give free aids
to the poor, for-profit firms have profit-making goals and SEs balance for-profit goals with social
objectives of creating better life for all. It is reported that SEs in the agricultural business expect
the procurement contracts to fight poverty and guarantee a reliable supply of agricultural products
(Global Envision 2012). For a SE, is there a contract that both improves farmers’ welfare and
brings enough profit to the SE to sustain its operations? For a profit-driven firm, is the goal of
profit maximization possibly compatible with improving farmers’ welfare? How are the answers to
these questions affected when farmers are heterogeneous in production costs and only a fraction of
farmers are able to participate in contract farming?
Building on the micro-foundation of farmers’ utility maximization, we answer the above questions
by considering farming contracts between farmers with heterogeneous production costs and a firm
(SE or profit-driven enterprise) that offers a procurement contract to a fraction of farmers. We
find that the optimal design of the contract is unique. Both farmers and the firm benefits from this
contract farming only if a unique procurement price is used. This price depends on the potential
size of the market and the farmers’ production costs. A lower price (i.e., insufficient subsidy) does
not alleviate the market fluctuation, while a higher price (i.e., over-subsidy) distorts the market
and hurts both the firm and some of the farmers. Moreover, we find that the optimal contract not
only brings equity in the long run among farmers of the same production cost, but also reduces
income disparity over time among farmers of heterogeneous costs, regardless of whether they are
contract farmers or not. This finding highlights the additional benefit of the win-win contract in
creating fairness.
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Overall our study contributes to the literature that addresses the challenges of reducing agricultural market fluctuations and improving farmers’ welfare in developing economies. More specifically, we first build a micro-foundation of farmers’ utility maximization, which makes possible
a detailed welfare analysis for farmers. Second, with the micro-foundation, we study interactions
among farmers with heterogeneous production costs in jointly determining the crop’s market price,
which in turn affects their welfare. We show that self-interested behavior by individual farmers can
be altruistic. Third, we design procurement contracts for social entrepreneurs or profit-driven firms
that are offered to a fraction of farmers and are win-win for both the firm and all farmers. In sum,
our results demonstrate that self-interested behavior by individual farmers, social entrepreneurs, or
profit-driven firms can be altruistic at the same time, benefiting all farmers and the whole society,
with the help of carefully disclosed information or optimally-designed contracts made available to
a fraction of farmers.
2.
Literature Review
Our paper is closely related to the stream of works on the cobweb phenomenon in the macroeconomics and agronomics literature. Ezekiel (1938) first attributes the cobweb phenomenon (recurring cyclical herding in production) to the fact that producers often base their short-run production
plans on the assumption that the present prices will continue. Our paper differs from that literature
in three main ways. First, our goal is to illustrate that the self-interested behavior of strategic farmers, social entrepreneur and for-profit firms can reduce the price fluctuations, whereas the focus
of the cobweb theory is to explain the price oscillations observed in various markets. Notably, we
take the perspective that such price oscillations can be mitigated by carefully designed, incentivecompatible mechanisms. Second, one key feature of our model is that we build a micro-foundation
of an individual farmer utility model with heterogeneous production costs, such that the supply
curve is derived from farmers’ utility maximization rather than being exogenously assumed as it
is in the cobweb models. With such a utility-based model, we can also evaluate each individual
farmer’s welfare. For example, we show that a carefully designed contract offered to a fraction of
farmers benefits all stakeholders and, at the same time, reduce income disparity among farmers.
Third, the cobweb models lead to the discussion about the rational expectations assumption and
its validity for all farmers (see, e.g., Nerlove 1958). We consider a fraction of farmers who have
rational expectations or who contract with social entrepreneurs or for-profit firms. This feature
emphasizes the strategic interactions among farmers who have different rationality or access to
external farming contracts. In a different context but a similar spirit, Su (2010) and Cachon and
6
Swinney (2009) consider the interactions between myopic and forward-looking customers, and Hu
et al. (2015a) consider the interactions among customers who have different granularity of information about the service system congestion. The latter shows that in the presence of information
cost, a fraction of customers may intentionally ignore the real-time information, thereby benefiting
social welfare. Aflaki et al. (2015) find that in the presence of hassle cost, a fraction of forwardlooking customers may intentionally choose not to search for more information and this rational
ignorant behavior may benefit the society as a whole.
As just mentioned, the paper is closely related to the strategic consumer literature. That is an
extensive stream of literature in operations management that considers the behavior of forwardlooking consumers in operational contexts (see, e.g., Su 2010, Cachon and Swinney 2011 and
references therein). In contrast to this stream of literature, we study the interactions between
forward-looking or firm-supported farmers and naı̈ve farmers, all serving as suppliers instead of
consumers. As in this literature, we resort to the concept of rational expectations equilibrium to
rationalize and characterize the behavior of forward-looking decision makers.
We also contribute to the economics and operations management literature that study “social
herding,” as our model captures the farmers’ shortsighted herding behavior in planting or abandoning crops. Veeraraghavan and Debo (2009) identify the herding effect in queues where consumers
infer service quality from the length of the queue. In our base model, herding is an outcome, not
a cause, of farmers’ shortsighted reactions to recent prices, and we study the role of self-serving
strategic farmers and firms in alleviating the herding in crop planting. We provide a solution under
which the herding effect can be eliminated by offering a carefully designed contract, which benefits both farmers and firms. Notably as opposed to the social herding effect, Tereyagoglu and
Veeraraghavan (2012), Agrawal et al. (2016) and Momot et al. (2016) consider consumers’ purchasing behavior that depends on their intrinsic desire for exclusivity. More broadly, we contribute
to this emerging literature on social operations management, which focuses on how social interactions leading to collective social behaviors (in this paper, individual farmers’ planting decisions
collectively determine the total production quantity and market price), have an impact on firms’
operational decisions (in this paper, procurement), and how operational decisions (in this paper,
e.g., contract farming) can influence the formation of collective social behaviors.
Our study also links to the literature showing that a lack of information can lead to endogenized
uncertainty in business practices. For example, the well-known “bullwhip effect” (see Lee et al.
1997) shows that when information about the true demand in the consumer market is missing,
the information transferred in a supply chain in the form of “orders” tends to be distorted and
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the variance of order quantities increases as one moves upstream along the supply chain. Hu et al.
(2015b) show that sales uncertainty can occur when customers have little idea about the true
value of a product and make purchasing decisions under social influence. They provide operational
and marketing policies for reducing such sales uncertainty, e.g., product variety reduction and
production postponement. Cooper et al. (2006) show that incorrect beliefs about customer behavior
cause a spiral-down effect, i.e., high-fare ticket sales, protection levels, and revenues systematically
decrease over time. In contrast, we study farmers’ crop planting decisions and find that shortsighted
reactions lead to boom-and-bust market prices and hurt farmers’ welfare. Such reaction may be
due to lack of sufficient information about the supply side including production costs, and about
the demand side including the potential market size and customers’ price sensitivity. We show that
providing sufficient information to some farmers to help them make rational decisions can deter
that undesirable situation. We further investigate win-win mechanisms that can be implemented
by a social entrepreneur or a for-profit firm to reduce the negative effects of the lack of market
information.
Poverty alleviation in developing countries has been studied in the burgeoning operations management literature that focuses on the design of socially responsible operations (see Sodhi and Tang
2014 for a comprehensive review). Many of those studies explore the value of disclosing market
information in helping farmers improve their welfare. Chen et al. (2013) study Indian conglomerate
ITC’s new business model, where the firm helps farmers to obtain various information through
E-Choupals network and allows farmers to sell directly to the firm at the market price. Chen and
Tang (2015) show that both private and public signals of the agricultural market reduce price
variations and improve farmers’ welfare. The value created by the private signal decreases in the
accuracy of the public signal; in the presence of the private signal, the public signal may not create
value. Chen et al. (2015) investigate the incentives for knowledge sharing among competing farmers. Farmers are allowed to ask questions on a digital platform and the questions are later answered
by some knowledgeable farmers or a representative expert. They find that the answers offered by
knowledgeable farmers are never more informative than those offered by the experts, and hiring
more experts induces the knowledgeable farmers to provide less informative answers. Tang et al.
(2015) examine whether under a Cournot competition farmers should utilize market information
to optimize their production plans when both the market potential and the process yield are uncertain. They find farmers will use market information in equilibrium, but their welfare may not be
improved unless the requisite upfront investment is sufficiently low. Belloni et al. (2016) show that
a monopsonistic buyer (a downstream principal in general) who has private demand information
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can contract with segmented farmers (upstream agents) to restore the first-best productive efficiency in an agricultural produce market even when each farmer has private information about his
own production capacity.
Our paper differs from those papers in that we focus on the scenario when the market information
is not available to all the farmers. We consider some farmers that can only react shortsightedly to
the market, and study how these farmers’ welfare is affected by self-serving strategic farmers and
firms. By modeling farmers’ dynamic crop planting decisions over time, we find that the self-serving
individuals and firms can also be altruistic and beneficial to all farmers. In terms of methodology,
we adopt a multi-period model with interactions both within and across periods, whereas all the
aforementioned papers build a one-period model.
Finally, our paper is also related to the literature on price protection mechanisms. Nicholls and
Opal (2005) review price protection implemented through fair trade operations. The effectiveness
of protection mechanisms is studied. Haight (2011) suggests that a fair-trade certification model,
which guarantees a fixed price if the farmers meet high production standards, can be poorly implemented and only applied to limited types of crops (Fair Trade USA 2012). Annan and Schlenker
(2015) study the effectiveness of federal crop insurance in the U.S. They argue that this government
program may be a disincentive for farmers to adapt to extreme weather. Most of the papers referred
to above are empirical studies. In our paper, we develop an analytical model to study the effectiveness of a pre-season procurement contract, offered by an SE or a for-profit firm, in reducing market
fluctuation and improving farmers’ welfare. We identify many potential risks associated with the
contract and find that a careful design is crucial to the effectiveness of such a contract. Federgruen
et al. (2016) consider a manufacturer that offers a menu of contracts to a set of contract farmers.
They focus on information asymmetry in farmers’ characteristics and develop efficient approaches
to designing the contracts. Mu et al. (2015) consider intermediaries that compete in collecting milk
from farmers. They find competition can either hurt or improve the quality of the milk, depending
on quality test mechanism. Lee et al. (2000) and Wang and Hu (2014) study the price adjustment
mechanism that aims to reduce price uncertainty but they study an entirely different setting from
this paper.
3.
Base Model with Only Naı̈ve Farmers
We study a multi-period model in which farmers make planting decisions for a single crop in each
period over an infinite horizon. The individual farmers are infinitesimally small-scale, and the size
of the farmer population is normalized to 1. At the beginning of period t, each farmer observes
9
pt−1 , the crop’s market price in the end of previous period t − 1 and decides whether or not to
cultivate the crop in the current period t. The planting decision depends on each farmer’s individual
assessment of the near-future crop price and the cost of planting the crop. The crop production
takes one full period. At the end of the period, all of the harvested crop is sold to the market with
the price pt determined by the total output from all farmers.
Farmers have heterogeneous endowment of production costs, which takes the form of c − δ where
c is a base constant and δ is uniformly distributed on [0, 1] capturing the idiosyncratic production
capabilities among farmers. Let F denote the cumulative distribution function (c.d.f.) of δ. Our
results would qualitatively hold for a generally distributed δ under generalized conditions. We first
assume all farmers are naı̈ve in that they react shortsightedly to the crop price when making
planting decisions. Specifically, naı̈ve farmers are incapable of correctly predicting the market price
and they simply take pt−1 from the last period as the indicator of the near-future price in the end
of period t. Therefore, a naı̈ve farmer’s perceived utility of planting the crop in period t, denoted
by unt , is
unt = pt−1 − (c − δ).
(1)
We premise that all farmers are unlikely to collaborate to make joint production decisions, given
their heterogeneity in production costs. In the beginning of each period, a naı̈ve farmer chooses
to cultivate the crop if and only if his perceived utility is no less than an outside option, which is
normalized to zero for all the farmers. We will extend our model by considering endogenized outside
options, the values of which are based on the farmers’ individual perceived utility of planting other
crops (see Section 7.1). We intend to use this base model to explain why an unstable market and
cyclical over- or under-production might occur. In Section 4 we will extend our model to account
for a fraction of strategic farmers who rationally anticipate the future price, and in Sections 5 and
6 we focus anew on social entrepreneurs and for-profit firms that (self-interestedly) induce naı̈ve
farmers not to make shortsighted decisions.
As farmers are infinitesimal and the size of the farmer population is scaled to 1, the total amount
of crop produced by farmers in period t, denoted as qt , is
qt = P r(unt ≥ 0) = 1 − F (c − pt−1 ).
(2)
As observed in many developing countries, it is not economical for small-scale farmers to store
or transport their crops, especially perishable crops, to distant markets. Thus, in our model we
assume all the crop is sold to a market to which all the farmers have access. We assume no yield
10
uncertainty for the base model and will relax this assumption in an extension (see Section 7.2).
Let Ω denote the size of the potential market demand. As farmers sell the homogeneous crop, the
market would be cleared at a market clearing price, associated with the total production quantity:
pt = Ω − qt = Ω − 1 + F (c − pt−1 ).
(3)
Note from (1)-(3) that farmers interact with one another through the market price. That is, the
collection of all individual farmers’ planting decisions determines the market price, which in turn
affects the realized utility of each one of them. In what follows, we will examine the connection
between farmers’ naı̈ve behavior and cyclical market fluctuations. For ease of exposition we make
the following assumptions throughout the paper.
Assumption 1 (Diversity in Planting Decisions). 0 < c − 1 < Ω < c + 1.
Assumption 2 (Regular Initial Price). (a) Ω − 1 ≤ p0 ≤ Ω and (b) c − 1 ≤ p0 ≤ c.
Assumption 1 rules out the trivial cases in which, no matter how the crop prices vary over time,
all farmers make the same production decisions in all periods. Specifically, from (1) and (3), it can
be easily shown that each farmer’s utility satisfies Ω − (c + 1) ≤ ut ≤ Ω − (c − 1). Thus, when the
market size is larger than c + 1, all farmers receive positive perceived utility and choose to plant
the crop in all periods. By the same token, all farmers choose to leave the market when the market
size is smaller than c − 1.
Assumption 2 regulates the initial market price. Assumption 2(a) is a natural assumption due
to the fact that pt = Ω − qt and 0 ≤ qt ≤ 1 for t ≥ 1. We could further narrow the range of p0 as
stated in Assumption 2(b) without loss of generality. That is because: (i) according to (2) and (3),
p0 influences the farmers’ crop production q1 and market price p1 only through F (c − p0 ); and (ii)
for any p0 < c − 1 we have F (c − p0 ) = F (1) = 1 and for any p0 > c we have F (c − p0 ) = F (0) = 0,
since F (·) is the c.d.f. of a random variable with support on [0, 1]. Hence, for all prices out of the
range of [c − 1, c], there exists an effective price on the boundary of the range that achieves the
same influence on the dynamics.
Now we use Eq.(1)-(3) to derive the market price evolution. Define
p̄ :=
Ω−1+c
.
2
Proposition 1 (Market Dynamics). The market price in period t is
pt = Ω − 1 + c − pt−1 = p̄ + (−1)t (p0 − p̄) .
(4)
That is, the market price alternates between two prices p2i = p0 and p2i+1 = 2p̄ − p0 for any i ≥ 0,
centered at p̄.
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Proposition 1 shows that the market price fluctuates considerably between two prices p0 and
2p̄ − p0 . It provides a micro-foundation of farmers’ utility maximization for the boom-and-bust
cobweb phenomenon: when the price realized at the end of a period is high, a very large number of
farmers flood to plant the crop in the upcoming period because farmers are all naı̈ve and think the
price in the next period will continue to be high. That results in over-production and a price drop
in the following period. Similarly, when the price realized at the end of a period is low, farmers tend
to abandon planting the crop (they may plant another crop, see Section 7.1 or seek other outside
options, such as going to the city as migrant workers), thus leading to a supply shortage and a
price rebound in the following period. The naı̈ve farmers get trapped in the vicious cycle and suffer
from the price fluctuations caused by the recurring under- or over-production. This unfavorable
situation will be manifested in the farmers’ welfare as follows.
Note that without market interferences, if p0 > p̄, then p1 < p̄, and vice versa, which indicates
the cyclical nature of a market with all naı̈ve farmers. Hence, without loss of generality (WLOG),
we assume:
Assumption 3 (WLOG). p0 > p̄.
All our results in the rest of the paper would hold for a system with p0 < p̄, when the possible
market interferences start from period 2.
A farmer’s individual welfare in period t is defined as the market price realized at the end of this
period minus the farmer’s production costs (i.e., pt − (c − δ)) if he plants the crop in period t, and
0 otherwise. Let market cycle i represent a pair of consecutive periods 2i and 2i + 1. Because of
the cyclical nature of the market price, a farmer’s welfare in different cycles is the same in the base
model. Denote w(δ) as the average welfare of a farmer with production costs c − δ in any cycle i.
Proposition 2 (Welfare). Consider a farmer with production costs c − δ in any cycle i.
(a) If c − δ ≤ 2p̄ − p0 , the farmer plants the crop in both periods and w(δ) = p̄ − (c − δ) > 0;
(b) If 2p̄ − p0 < c − δ ≤ p0 , the farmer plants the crop only in period 2i + 1 and w(δ) = 21 [2p̄ − p0 −
(c − δ)] < 0;
(c) If p0 < c − δ, the farmer does not plant the crop and w(δ) = 0.
The micro-foundation allows us to perform a detailed welfare analysis for farmers beyond the
cobweb theory. Proposition 2 tells us that farmers’ welfare depends on their production costs. It
is noticeable that only farmers with very low costs (i.e., c − δ ≤ 2p̄ − p0 ) or very high costs (i.e.,
p0 < c − δ) are not directly affected by market fluctuation. The low-cost farmer plants the crop
in all periods as his cost is always below the market price; the high-cost farmer does not plant
12
Figure 2
A naı̈ve farmer’s revenue with 2p̄ − p0 < c − δ ≤ p0 .
Market Price
−
2 −
$
$
0
1
2
3
$
4
5
$
$
6
$ A naıve farmer’s revenue
7
8
9
$
10
11
12
Periods
Market price
Note. The naı̈ve farmer with an intermediate cost level (i.e., 2p̄ − p0 < c − δ ≤ p0 ) chooses to plant the crop only
when the most recent price is high, i.e., he plants in odd periods. Therefore, the farmer receives revenues only in odd
periods that are unfortunately below his production cost, leading to negative welfare, as shown in Proposition 2(b).
the crop in any period as his production cost always surpasses the market price. Unfortunately, as
Figure 2 illustrates, the majority of the farmers who have intermediate cost levels (i.e., 2p̄ − p0 <
c − δ ≤ p0 ) are hurt by their naı̈ve strategy, i.e., planting the crop when the latest crop price is high
and abandoning the crop when the price is low. Their individual welfare is negative as shown in
Proposition 2(b). On top of the base model, we will examine the way in which strategic farmers
and social entrepreneurs or for-profit organizations influence the farmers’ planting decision and
possibly enhance their welfare.
4.
Strategic Farmers
In this section, we consider a situation where a fraction, α, of the farmers is strategic and the
remaining 1 − α fraction remains naı̈ve. To anticipate the near-future market price, strategic farmers
need (i) to have full information about their fraction α, the market potential Ω, and production
costs distribution F (·) and (ii) to take rational expectation to predict the near-future market price.
Assumptions (i) and (ii) are widely used in the literature to model strategic consumer behavior
(see, e.g., Cachon and Swinney 2009). We use these assumptions to exclude any tactic behavior
that is imperfect and to examine the influence of strategic farmers on market dynamics. Shortly
we will also discuss the situations where these assumptions are relaxed.
At the beginning of period t, anticipating the near-future market price pt , a strategic farmer’s
perceived utility of planting the crop is
ust = pt − (c − δ).
(5)
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The shortsighted behavior of naı̈ve farmers remains the same as specified in Section 3. They will
plant the crop only if their individual perceived utility is nonnegative. Thus, the quantities produced
by all strategic farmers and naı̈ve farmers, denoted as qts and qtn , are
qts = αP r(ust ≥ 0) = α[1 − F (c − pt )],
(6)
qtn = (1 − α)P r(unt ≥ 0) = (1 − α)[1 − F (c − pt−1 )].
(7)
The market clearing price associated with the total production quantity is
pt = Ω − qtn − qts
(8)
= Ω − 1 + αF (c − pt ) + (1 − α)F (c − pt−1 ).
(9)
Note that pt appears in both sides of the above equality, implying that the strategic farmers
anticipate the future price rationally when making planting decisions. Now we use (9) to derive
the market price evolution.
Proposition 3 (Market Dynamics). If there exist strategic farmers (α > 0), the market
price in period t is
t
1−α
Ω − 1 + c − (1 − α)pt−1
= p̄ + −
pt =
(p0 − p̄) .
1+α
1+α
(10)
The market price pt converges to p̄, i.e.,
lim pt = p̄.
t→∞
That is, even a tiny number of strategic farmers can stabilize the market price.
Proposition 3 shows that the market price is a weighted sum of two values, p̄ and (p0 − p̄), with
t
on the latter. Note that this weight alternates between positive
a time-varying weight − 1−α
1+α
and negative values, and its absolute value decreases, which imply pt varies around p̄ over time
and eventually converges to p̄ independently of the initial price. Thus, we name p̄ as the limiting
market price.
The strategic farmers can rationally anticipate market prices and avoid herding with naı̈ve
farmers. Surprisingly, we find that their self-serving planting decisions always help stabilize market
prices in the long run, even if there is only a tiny number of strategic farmers in the population.
That is because the planting decisions by forward-looking rational farmers can gradually balance
the market output. Moreover, the fraction of strategic farmers α affects only the convergence rate.
14
Corollary 1. The fraction of strategic farmers α affects only the convergence speed of the
market price but not the limiting market price. The larger the fraction of strategic farmers α is,
the faster the market price converges.
The following proposition examines the role of strategic farmers in enhancing farmers’ welfare.
Proposition 4 (Altruistic Rationality). In each period, a strategic farmer earns a higher
welfare than a naı̈ve farmer with the same production cost. Moreover, each naı̈ve farmer’s welfare
is higher than it is in the absence of strategic farmers.
Proposition 4 highlights the importance of cultivating strategic farmers. The first part of it shows
that the forward-looking behavior is indeed rational, and the second part shows that such strategic
farmers’ rational behavior is also altruistic and helps increase the welfare of naı̈ve farmers. The
welfare improvement for naı̈ve farmers comes from the stabilized market price that results from
the presence of strategic farmers.
Although there is benefit in being forward-looking, farmers may lack the intrinsic reasoning
ability or the external aids to obtain full market information and may thus fail to make fully
rational decisions. In the rest of the section, we will explicitly incorporate such farmers’ limitations
in our model and show the impact of imperfect strategic behavior on the market.
4.1.
Backward Looking
In this extension, we assume farmers lack full information (e.g., Ω, α and F (·)) and thus cannot
use (9) to rationally anticipate the near-future market price pt . Instead, in each period strategic
farmers predict the market price on the basis of the average price in the last two periods, i.e.,
p̂t =
pt−1 +pt−2
.
2
This is a simple but widely used adaptive forecasting method (see Chopra and
Meindl 2013, Chapter 7), as it is easier to obtain the necessary information about historical prices
than future prices. The strategic farmers’ perceived utility becomes
ust = p̂t − (c − δ) =
pt−1 + pt−2
− (c − δ).
2
(11)
We adopt the convention that pi = p0 for any i < 0. By the same decision and price processes as in
(6)-(8) with pt replaced by p̂t , for any t ≥ 1, we have
pt = Ω − 1 + (1 − α)F (c − pt−1 ) + αF (c −
pt−1 + pt−2
).
2
(12)
Proposition 5. If there exist backward-looking strategic farmers (α > 0), then the market price
converges to the limiting market price, i.e.,
lim pt = p̄.
t→∞
15
Proposition 5 tells us that as long as there are backward-looking strategic farmers, the market
price converges to the same value as if the strategic farmers had full information and were forwardlooking. More importantly, it implies that looking into the past can be as efficient as looking into
the future in a stationary market.
4.2.
Bounded Rationality
Unlike in the previous extension, we now assume that strategic farmers have full market information but bounded rationality in making planting decisions. The bounded rationality essentially
recognizes the individual’s imperfect discriminatory power and insufficient ability to process information. When faced with a choice among multiple options, one is likely to exhibit uncertainty and
inconsistency (see Anderson et al. 1992 for a review). Accordingly, we view the strategic farmers’
planting choice as a probabilistic decision-making process. The strategic farmers’ perceived utility
of planting the crop ust stays the same as (5), and the utility of not planting the crop is 0. Like Su
(2008), we use a multinomial logit model to specify the probability denoted by ϕt that a strategic
farmer chooses to plant the crop, i.e.,
s
ϕt (δ) =
e(pt −c+δ)/β
eut /β
,
s /β =
u
1 + e(pt −c+δ)/β
e0/β + e t
where β represents the extent of bounded rationality. The probability of not planting is 1 − ϕt (δ).
As β → ∞, ϕt → 1/2 corresponds to a random selection between planting and no planting. As
β → 0, ϕt → 1 if ust ≥ 0 and ϕt → 0 otherwise, corresponding to a fully rational (utility-maximizing)
decision. For tractability, we take ϕt (δ) as the proportion of strategic farmers who choose to plant
the crop and have production cost c − δ. The total quantity produced by strategic farmers is
qts
Z
=α
1
1
Z
ϕt (δ)dδ = α
0
0
1 + e(pt −c+1)/β
e(pt −c+δ)/β
dδ
=
αβ
ln
.
1 + e(pt −c+δ)/β
1 + e(pt −c)/β
Together with (7) and (8), we obtain the market clearing price:
pt = Ω − (1 − α)[1 − F (c − pt−1 )] − αβ ln
1 + e(pt −c+1)/β
.
1 + e(pt −c)/β
(13)
Proposition 6. Suppose that there exist strategic farmers (α > 0) with bounded rationality and
that c − 1/2 ≤ Ω ≤ c + 1/2. Then, the market price converges to a unique price X, where X is
characterized by the equation
X + αβ ln
1 + e(X−c+1)/β
= Ω − (1 − α)(1 − c + X).
1 + e(X−c)/β
16
Proposition 6 shows that the market price still converges even if the strategic farmers have
bounded rationality. The limiting market price depends on the extent of bounded rationality.
However, we find that if the strategic farmers are not sufficiently rational (i.e., β is sufficiently
large), the farmers may become worse off. In the online appendix (see Lemma OA.4), we show
that in the extreme case when β reaches infinity, both naı̈ve and strategic farmers may obtain even
lower welfare than that in the base model where all farmers are naı̈ve. Moreover, in this extreme
case when β reaches infinity, when the market size is small (e.g., Ω ≤ c), all farmers suffer lower
welfare, as compared to the counterpart with strategic farmers being fully rational (i.e., β = 0).
That is, bounded rationality can result in welfare loss for farmers, although it still leads to price
stabilization. Therefore, it is imperative to discover other ways of helping farmers more effectively.
5.
Social Entrepreneurs
In the last section, we showed that the sheer existence of the strategic farmers, even in a tiny
amount, would be sufficient to tilt the market dynamics from fluctuation to long-term stabilization. However, it is generally observed that it has been very difficult to cultivate enough strategic
farmers to speed up the market stabilizing process for several reasons. First, in the case of fresh,
local specialty fruits and vegetables, there is often a lack of extensive crop advisory systems that
would help farmers obtain the market information about each crop and track the activities of other
farmers. Second, farmers’ limited ability to predict crop prices and their shortsighted behavior
have remained major difficulties despite considerable advances in crop advisory communications.
Moreover, external factors such as market uncertainty may add significant complexity to the market stabilizing efforts, as we will discuss later in Section 7. For those reasons, in many developing
countries where those problems are common, it may not be as effective as it sounds to rely purely
on farmers’ forward-looking behavior to stabilize the market. What is called for is a market stabilization effort by social organizations who can operate in a sustainable way.
In this section, we consider a social entrepreneur (SE) who focuses on both the benefits of the
society and the profit gains to sustain his own operations. The SE subsidizes a fraction of farmers
by offering them a pre-season procurement contract that buys out all the crop from them, when
naı̈ve farmers do not voluntarily become strategic. We refer to the pre-season procurement contract
as a buyout contract. In this contract SE selects and announces a fixed purchase or buyout price
pb for all periods, and at the beginning of each period each farmer who is offered the contract
decides whether to sign the contract with the SE. (We will discuss time-dependent buyout price
in Subsection 5.3.) Farmers who accepted the SE’s contract at the beginning of period t sell their
17
crops to the SE at the price pb at the end of period t. Right after the SE buys the crops from the
farmers, he immediately sells them on the market at the market price pt . The pre-season buyout
contract we study here is different from the minimum-price support program offered by an NGO
or government. In the latter, the NGO or government is required to buy the crop when prices fall
below a set level after the harvest, while our contract is agreed on before the season starts. The
pre-season contract can help guide the planting decisions in the first place, whereas the minimumprice support program may have a delayed effect. Though the program can also guarantee farmers’
payoffs, it may not effectively guide planting decisions, and very likely comes at the cost of NGOs
or governments (see, e.g., Wall Street Journal 2016b). Unlike Annan and Schlenker (2015), we
assume away moral hazard issues in minimum-price support program or contract farming. In the
following subsection we will examine whether the aid from the SE would bring sustainable benefits
to all farmers and to the SE himself as well.
5.1.
Market Equilibrium in the Presence of SE
We assume that all farmers are naı̈ve and only an exogenous, α, fraction of farmers have access to
the SE’s contract. That could be attributed to many factors. For instance, some farmers in rural
areas may be too remote to be reached and others may decline to work with an unfamiliar SE.
We define farmers who are offered the SE’s contract as type-S farmers and farmers who are not as
type-N farmers. A type-S farmer’s perceived utility, if he accepts the contract and plants the crop,
is pb − (c − δ). Since the farmer would otherwise still behave naı̈vely, he will accept the contract
only if the offered price pb is strictly higher than the market price pt−1 in the last period. Hence,
a type-S farmer’s perceived utility in period t, denoted by ubt (where b corresponds to the buyout
contract), is
ubt =
pb − (c − δ) for pb > pt−1 ,
pt−1 − (c − δ) otherwise.
(14)
In other words, farmers perceive to be incentive-compatible in deciding whether to accept the
buyout contract. The SE’s buyout contract aims to stimulate farmers to plant crops in a period
when the market price in the last period was low. The contract can also be viewed as a crop
insurance since it is evoked only contingently, i.e., when the most recent market is down. The
corresponding quantity produced by type-S farmers, denoted by qtb , is
qtb = αP r(ubt ≥ 0).
(15)
The perceived utility of the rest 1 − α fraction of type-N farmers unt and their production quantity
qtn remain the same as (1) and (7), respectively. The market clearing price in period t becomes
pt = Ω − qtb − qtn
(16)
18
=
Ω − 1 + αF (c − pb ) + (1 − α)F (c − pt−1 ) for pb > pt−1 ,
Ω − 1 + F (c − pt−1 )
otherwise.
(17)
As in Assumption 2(b), it can be easily shown that pb makes a difference to the problem only
when it falls in the range specified in the following assumption.
Assumption 4. c − 1 ≤ pb ≤ c.
We now study the long-term market trends for a certain level of buyout price pb . According to
Proposition 1, if there are no strategic farmers and no contract offered by the SE, the market price
fluctuates up and down over time, centered around the limiting market price p̄. The SE’s contract
would stimulate planting the crop when the market price is below pb and thus reduce the price
fluctuation. The following proposition shows that the extent to which the SE’s buyout contract
can help reduce the price fluctuation varies, depending on the buyout price pb and the market size
Ω.
Proposition 7. If pb < p̄, the market price fluctuates between two prices without convergence.
If pb ≥ p̄, then the market price converges. Specifically, denoting p̊ :=
Ω−(1−α)(c−1)
α
(a) If (i) pb > p̊ and Ω ≥ c or (ii) p̄ ≤ pb ≤ p̊, then the market price converges to
> p̄, we have:
Ω−1+c−αpb
.
2−α
(b) If pb > p̊ and Ω < c, then the market price becomes Ω + α (c − 1 − pb ) for any t ≥ 2. All type-N
farmers (who are not offered the contract) do not cultivate the crop.
As stated in Proposition 7, if the buyout price pb is below p̄, then the SE’s contract does not
stabilize the crop price. That is because by the time the price fluctuates above pb , no farmers
would accept the buyout contract and thus the contract is no longer effective and the price will
not continue to converge. In this situation, farmers are still exposed to the price risks.
When pb is greater than the limiting market price p̄, the market price converges. In the online
appendix, the proofs of Lemmas OA.5 and OA.6 demonstrate the path of the price convergence over
time. In particular, when pb is moderate, or when it is high and the market size is large, Proposition
7(a) shows that the market price converges to
Ω−1+c−αpb
2−α
≤ pb , implying SE may incur losses from
his operations. When pb is high and the market size is small, Proposition 7(b) shows the market
prices quickly converge to a very low price Ω + α (c − 1 − pb ) < c − 1, resulting in negative perceived
utility for type-N farmers, i.e., unt < 0. In such a situation, the high buyout price induces more
type-S farmers to produce the crop and then dump their surplus on the local market, crash the
price and crowd out all type-N farmers, i.e., the naı̈ve farmers who are not offered the contract. This
market dynamics is exemplified by many charitable programs whose unfair subsidies undermine
the efforts of the poor and thus may end up with sustaining their poverty (see Miller 2014). In
summary, we have:
19
Corollary 2. An inadequate subsidy (i.e., low buyout price) does not eliminate the fluctuations
in market price. Too large a subsidy (i.e., a high buyout price), however, may create huge distortions
in the local market, cause many non-contract farmers to give up farming, and result in losses for
the SE.
Corollary 2 reveals that an SE’s efforts, which are inspired by a desire to do good, do not always
have a satisfactory outcome. A buyout subsidy that is too high benefits contract farmers at the
expense of non-contract farmers and may actually create more poverty. This should be a caveat
for governments and NGOs that may have more centralized power to provide subsidies, perhaps
in the name of charity. Therefore, it is desirable to find the optimal design of the buyout contract.
5.2.
Optimal Design of Buyout Contract with Multiple Goals
As mentioned above, the SE’s goal is to stabilize the market while generating enough profit to
sustain his operations. The SE’s profit in period t depends on pb and the market prices in periods
t − 1 and t. According to (14), if pb > pt−1 , then all type-S farmers with production cost c − δ < pb
accept the buyout contract. Thus, the SE buys crops from type-S farmers in period t at unit price
pb and then sells them at unit price pt ; thus the SE’s profit is (pt − pb )qtb . If pb ≤ pt−1 , then no
type-S farmers accept the buyout contract in period t and thus the SE’s profit in period t is 0.
As stated in Proposition 7, if pb < p̄, the market is unstable; if pb > p̄, the converged market price
is less than pb and therefore the SE incurs losses. Hence, setting pb equal to the limiting market
price p̄ is the only way for SE to stabilize the market price through the buyout contract and at
the same time avoid any loss in the long run. In the rest of this subsection we study the market
dynamics, the SE’s profit, and the farmers’ welfare under the buyout contract with pb = p̄, which
is the optimal fixed buyout price, if the SE’s goal is to stabilize the market and benefit farmers,
while sustaining his operations profitably.
Proposition 8 (Market Dynamics). Suppose SE sets the price at pb = p̄. The type-S farmers
accept the contract only in even periods and the corresponding market price in cycle i is
i
p2i = p̄ + (1 − α) (p0 − p̄) ,
p2i+1 = p̄ − (p2i − p̄) ,
(18)
with p2i+1 < p̄ < p2i and limi→∞ p2i = limi→∞ p2i+1 = p̄.
According to Proposition 8, the SE contracts with farmers only in even periods and the correb
sponding profit gain in cycle i is (p2i − p̄)q2i
= (1 − α)i (p0 − p̄)α[1 − F (c − p̄)] > 0, meaning that the
SE actually profits over any finite horizon but his profit falls in each successive cycle.
20
Now it is tempting to see farmers benefit from this contract. Let w̄is (δ) and w̄in (δ) be the average
welfare of, respectively, a type-S farmer and a type-N farmer with production cost c − δ in cycle i.
Specifically, we have
w̄is (δ) =
o
o
1n
1n
s
s
n
n
1{ub ≥0} w2i
+ 1{ub ≥0} w2i+1
and w̄in (δ) =
1{un ≥0} w2i
+ 1{un ≥0} w2i+1
,
2i
2i+1
2i
2i+1
2
2
where wts and wtn denote the welfare of, respectively, a type-S and a type-N farmer in period t if
he plants the crop, and where 1{A} is an indicator function that equals 1 if condition A holds and
otherwise equals 0. The indicator function is used to specify the criteria for planting the crop, i.e.,
the perceived utility is nonnegative. In the online appendix (see the proof of Lemma 4), we fully
characterize the functions w̄is (δ) and w̄in (δ). The average welfare of a farmer in any cycle i when
the SE is absent, w(δ), has been given in Proposition 2. In the following we show the impact of
the SE’s contract on farmers’ welfare by comparing w(δ) with w̄is (δ) and w̄in (δ).
Proposition 9 (Long-Run Equity Among Farmers of the Same Cost). Suppose
the
SE sets the price at pb = p̄. Consider a farmer with production cost c − δ. As i approaches ∞,
(a) if c − δ ≤ p̄, the farmer plants crops in both periods and limi→∞ w̄is (δ) = limi→∞ w̄in (δ) = p̄ −
(c − δ);
(b) if p̄ < c − δ, the farmer does not plant the crop and limi→∞ w̄is (δ) = limi→∞ w̄in (δ) = 0.
In the limit, the market price converges to p̄ (see Proposition 8), which is the same as the buyout
price offered by the SE. Thus, a farmer regardless of his type, plants the crop in every period
in the limit if his production cost is lower than p̄ and otherwise does not plant in any period.
Accordingly, as shown in Proposition 9, both type-S farmers and type-N farmers receive the same
average welfare in the long run, i.e., limi→∞ w̄is (δ) = limi→∞ w̄in (δ). Denote ∆(δ) as the long-run
farmers’ welfare improvement introduced by the SE. This value is the same for type-S and type-N
customers in the long run. That is, ∆(δ) := limi→∞ w̄is (δ) − w(δ) = limi→∞ w̄in (δ) − w(δ). Based
on Propositions 2 and 9, we find farmers benefit from SE and more importantly, the distribution
of benefit is fair in the long run among all farmers with the same production cost, regardless of
whether a farmer deals with SE directly (i.e., ∆(δ) ≥ 0). The welfare improvement comes from
stabilizing the market price to avoid farmers’ cyclical over- and under-production.
However, the welfare improvement ∆(δ) is not monotone in a farmer’s cost endowment c − δ. If
a farmer’s production cost is low (i.e., c − δ ≤ 2p̄ − p0 ) or high (i.e., p0 < c − δ), there is no welfare
change (i.e., ∆(δ) = 0). If a farmer’s production cost is medium-low (i.e., 2p̄ − p0 < c − δ ≤ p̄),
because of the SE, the farmer switches from planting crops in odd periods in an unstable market
21
(Proposition 2(b)) to planting crops in all periods in a stable market (Proposition 9(a)). Thus,
the farmer’s welfare increment is ∆(δ) = p0 − c−δ
, which is positive and implies that farmers with
2
higher production costs within the medium-low cost band see less welfare improvement. On the
contrary, if a farmer’s production is medium-high (i.e., p̄ < c − δ ≤ p0 ), he switches from planting in
a single period (Proposition 2(b)) to not planting at all (Proposition 9(b)). His welfare improvement
is ∆(δ) =
c−δ
2
−
2p̄−p0
,
2
which is positive and implies that farmers with higher production costs
within the medium-high cost band avoid more losses and thus see larger welfare improvement. The
proposition below summarizes the impact of the buyout contract pb = p̄.
Proposition 10 (Altruistic Rationality). The SE is able to stabilize the market price
without a profit loss only if the buyout price is set at pb = p̄. In that case:
(a) The SE receives a positive profit over any finite horizon and the profit per market cycle diminishes to 0 in the limit.
(b) Both type-S and type-N farmers receive (weakly) higher welfare in the long run due to the
market stabilization in the presence of SE. The incremental welfare is not monotone in farmers’
cost endowment c − δ.
Propositions 7 and 10 hint that price protection by SE can be a precarious business model and
that in our deterministic and stationary world, there is a unique design of the buyout contract that
leads to a win-win situation for the SE and the farmers in the long run. Moreover, we find that in
each period this optimal buyout contract reduces the welfare gap between farmers who are at an
advantage because of their production costs and those who are at a disadvantage.
Proposition 11 (Fairness Across Classes of Farmers with Heterogeneous Costs).
Under the optimal buyout contract pb = p̄, the income disparity between low-cost farmers and the
other farmers is reduced. More specifically, for any given δ1 and δ2 such that c − δ1 ≤ 2p̄ − p0 < c − δ2 ,
for any cycle i,
w̄is (δ1 ) − w̄is (δ2 ) ≤ w̄in (δ1 ) − w̄in (δ2 ) ≤ w(δ1 ) − w(δ2 ).
The quantity 2p̄ − p0 equals to the lowest market price. Farmers whose production costs c − δ1
are below 2p̄ − p0 have a significant cost advantage over the rest of the farmers. Proposition 11
shows that with the optimal buyout contract of an SE, the gap between high- and low-cost farmers
is smaller among type-S farmers than that among type-N farmers, which is further smaller than
that among farmers in the scenario with all naı̈ve farmers. Figure 3 illustrates farmers’ revenue
and welfare (or income, i.e., revenue minus production cost) in cycle i. The farmers are categorized
by their type and production cost. As displayed in Figure 3(b) and shown in Proposition 11,
22
Figure 3
Revenue and welfare in cycle i with c − δ1 < 2p̄ − p0 < c − δ2 < p̄.
(a) Revenue
(b) Welfare
$
̅
$$
−
$$$ $
−
0
2 −1
2
$
$
$
$
2 +1
2
2 +1
Type-S farmer with production cost −
Type-S farmer with production cost −
Type-N farmer with production cost −
Type-N farmer with production cost −
Note. In period 2i both type-S farmers with production cost c − δ1 and c − δ2 choose to contract with SE and plant
the crop, because the contract price p̄ is higher than their production cost and the most recent price p2i−1 . Only
type-N farmers with production cost c − δ1 < p2i−1 choose to plant the crop. The legends in (a) indicate farmers’
corresponding revenues. Similarly, due to the high price p2i , all farmers plant the crop in period 2i + 1 without
contracting with SE and they receive the same revenue that equals the market price p2i+1 .
the welfare difference among type-S farmers with high and low production costs is less than that
among type-N farmers. Therefore, SE not only helps improve total welfare but also helps reduce
the income gap between the advantaged and disadvantaged farmers. That is, the distribution of
benefit also tends to be fair across farmers with heterogeneous production costs, in addition to its
long-run fairness among farmers with the same production cost, regardless of whether they are
contract farmers or not.
5.3.
Buyout Contract Implementation
There are several difficulties in implementing the buyout contract. First, as shown in Lemma 4(a)
(in the appendix), before the market price converges, the low-cost farmers who are offered the
contract periodically see the market price that is higher than the contract price pb at which they
agreed to sell to the SE. They may realize that their average welfare is lower than it was in the
past when there was no SE, i.e., w̄is (δ) < w(δ). Thus, these farmers may choose to default on the
contract with the SE before the market stabilizes. However, our analysis implies that their possible
default has no impact on the crop supply and hence none on the market price, because these
low-cost farmers will plant the crop in both periods regardless of whether they decide to sell on
the market or to the SE. Thus, contract price adjustment or reward in any other form to prevent
these low-cost farmers from possibly defaulting is not necessary.
23
Second, since the SE’s contract is accepted by farmers only when the market price is low, it
is effective only in one period of every market cycle. Thus, it can be shown that the convergence
rate of the market price is lower than that when the convergence results from the forward-looking
behavior of strategic farmers in Section 4. Formally, we have:
Proposition 12. Given the same α, the market price in the model with the social entrepreneur’s
contract converges more slowly than that in the model with strategic farmers.
This indicates that cultivating strategic farmers can be a more effective strategy over the long
run than simply offering naı̈ve farmers a buyout contract.
Third, depending on the weight on social welfare in the SE’s objective, the firm could be more
speculative by offering a dynamic buyout contract. The SE could adjust the buyout price pb dynamically in order to earn higher profits in the short run. For instance, based on (14), when pt−1 is low
(which leads to less production), the SE could contract with farmers at a price even lower than p̄
but strictly higher than pt−1 , and sell at a higher market price in period t. That could dramatically reduce the rate of the market convergence and result in a lower farmers’ welfare. Therefore,
monitoring and regulation of SE’s behavior can be necessary.
6.
Profit-Driven Enterprise
While the important role of SEs and other social organizations in increasing farmers’ welfare
have been widely acknowledged, one cannot expect their participation in every kind of agriculture
industry. In this section, we further study the most common interactions between farmers and a
profit-driven supplier who buys raw materials from upstream farmers for resale or may use these
materials to make products for sale. When all the farmers are naı̈ve, we have seen that the price
uncertainty is a major obstacle to a stable stream of crop supply. In this situation, reducing the
price risks and having a stable market supply may benefit all the stages along the supply chain.
Therefore, it is often in the best interest of the supplier to plan ahead to contract with farmers to
ensure a safe and steady supply of agricultural products. These companies may not want to wait
for actions taken by other organizations. Hence, it is important to examine how the crop supply
risk and market price risk can be mitigated solely by a profit-driven firm. Specifically, we study
the impact of the buyout contract when it is offered by the profit-driven firm instead of an SE.
Suppose the firm has a constant demand d in each period. This demand d can also be generally
interpreted as a firm’s target level of product supply. Assume that one unit of raw materials can
be converted to one unit of products. If the market supply qt > d, the firm buys d units of raw
materials from the market. If qt < d, the firm has to buy additional quantity d − qt from an external
24
market at a much higher price pm with pm > pt for any t ≥ 0. The external market is isolated from
the focal market we have been studied. The difference pm − pt can also be viewed as a penalty cost
for those demands in excess of supply. We start with all farmers in the population being naı̈ve
farmers. Just like the SE in the previous section, the firm offers a pre-season procurement contract
at the beginning of each period with a fixed price pb to α fraction of naı̈ve farmers. As in the
previous section, we refer to the farmers who are offered the contract as type-S farmers and the rest
as type-N farmers. In order to focus our study on scenarios where a stable market has a significant
impact on the firm’s profit, we impose the following assumptions about the firm’s demand d.
Assumption 5. (a) Ω − p0 < d < Ω − p̄ and (b) α(Ω − p̄) < d.
Assumption 5(a) ensures that a stable market can provide sufficient supply to meet the firm’s
demand, whereas an unstable market may not. Recall that when there are only naı̈ve farmers
without contracts, according to Proposition 1 the market price fluctuates between two prices p0
and Ω − 1 + c − p0 (< p0 ). Assumption 5(a) indicates the crop supply at the initial market price p0 ,
i.e., Ω − p0 , is less than the firm’s demand d. However, when the firm offers buyout contracts with
price pb = p̄, according to Proposition 8 the market price is stabilized at p̄ and the corresponding
crop supply is Ω − p̄, which is greater than the firm’s needs d as specified in Assumption 5(a).
Assumption 5(b) further states that the production by type-S farmers α(Ω − p̄) alone is insufficient
to satisfy all of the firm’s demand. Hence, the conditions in Assumption 5 ensure that the firm
hopes for a stable market, and has an incentive to induce both types of farmers to produce enough
of the crop for the firm.
We now study the performance of the buyout contract with pb = p̄ when it is offered by the
profit-driven firm instead of the SE. Note that the market evolution and farmers’ welfare under
the contract should be the same as specified in Propositions 8 and 10(b).
Let f n and f i be the firm’s average procurement cost in cycle i when the firm buys directly
from the market without offering a buyout contract and when it offers the buyout contract with
pb = p̄ respectively. If the firm does not offer any buyout contract, the market price would alternate
between two prices p0 and Ω − 1 + c − p0 . The average procurement cost per cycle is
fn =
1
{p0 (Ω − p0 ) + pm [d − (Ω − p0 )] + (Ω − 1 + c − p0 ) d} .
2
Note here that the crop supply is insufficient when the market price is p0 , but is sufficient to meet
demand d when the market price is Ω − 1 + c − p0 < p̄ according to Assumption 5(a).
Next we derive the market supply under the buyout contract and the corresponding f i , and then
compare it with f n . With the buyout contract pb = p̄, Proposition 8 shows that the market price
25
fluctuates around p̄ with shinking ranges, i.e., p2i (> p̄) decreases in i and p2i+1 (< p̄) increases in
i. According to (17), the realized market price in a period is determined by the underlying crop
supply quantity in an inverse relationship. As the cycle index i increases, the market price in period
2i decreases from p0 to p̄, which implies the crop supply increases from Ω − p0 to Ω − p̄. Thus, for
demand d that satisfies Assumption 5(a), there exists a threshold on the cycle index,
i0 := min{i | p2i ≤ Ω − d},
such that the market supply is sufficient to meet demand d if and only if i ≥ i0 . In period 2i + 1 for
any i ≥ 1, since the market price satisfies p2i+1 < p̄, the crop supply is greater than d according to
Assumption 5(a).
The procurement cost depends not only on the market supply qt but also on whether the firm
buys through the buyout contract or from the market directly. Let qtb and qtn be the market supply
in period t that is provided by type-S and type-N farmers respectively. We have qt = qtb + qtn .
b b
p to buy crops from
Consider the procurement cost in cycle i < i0 . In period 2i, the firm pays q2i
type-S farmers since they all accept the buyout contract due to p2i−1 < pb , pays the market price
b
n
p2i to buy all crops provided by type-N farmers, and pays (d − q2i
− q2i
)pm to buy the rest of crops
the firm needs from the external market. In period 2i + 1, no type-S farmers accept the buyout
contract since p2i > pb and the firm procures crops at the market price p2i+1 for d units because the
market has ample supply in those periods. The procurement cost in cycle i ≥ i0 can be easily found
in the same way. To sum up, with the buyout contract pb = p̄, the firm’s average procurement cost
in market cycle i, denoted by f i , is
1 b b
n
b
n
p q2i + p2i q2i
+ pm (d − q2i
−q2i
) + p2i+1 d for i < i0
i
2
f = 1 b b
b
p q2i + p2i (d − q2i
otherwise.
) + p2i+1 d
2
The following proposition compares the firm’s procurement cost when it offers a buyout contract
with that when it does not.
Proposition 13. (a) If i > i0 or Ω − 2p0 + pm − d > 0, then f i < f n . That is, the cost per cycle
under the contract with pb = p̄ is strictly lower than that without a contract.
(b) limi→∞ f i is the lower bound of the procurement cost in a cycle under all possible buyout
contracts with fixed buyout price that can stabilize the market. That is, the fixed term pb = p̄ is the
optimal contract under the long-run average criteria.
Proposition 13(a) shows that the provision of a buyout contract with pb = p̄ benefits a profitdriven firm in both the short term (as long as Ω − 2p0 + pm − d > 0 holds) and the long term for
26
i > i0 . The condition Ω − 2p0 + pm − d > 0 holds if and only if f n increases in p0 . Since the range of
the price fluctuation 2(p0 − p̄) (see Proposition 3) is linear in p0 , this condition essentially means the
procurement cost increases in the price volatility, a finding which is in line with many observations
in practice. Moreover, Proposition 13(b) shows that the buyout contract with pb = p̄ is optimal
among all buyout contracts with fixed contract price pb under the long-run average procurement
cost criteria. Since this contract also increases the farmers’ welfare in the long run (see Proposition
10(b)), it can be seen that a profit-driven firm’s selfish behavior can also benefit all farmers.
7.
Extensions
The unstable market supply that results from the price fluctuations is often made worse by a host
of endogenous and exogenous causes. For example, farmers may choose what crop to plant among
multiple crops based on their market prices, and this decision can be influenced by crop production
yield, which is affected by weather, insect pests, etc. In this section, we first extend Section 4 to a
two-crop setting in which both naı̈ve and strategic farmers choose to cultivate one of two different
types of crops. In other words, the outside option of planting one crop for a farmer is endogenized as
the value of planting the other crop. We then further extend this model to capture the production
yield uncertainty. The same extensions can be naturally added to the settings in Sections 5 and
6. Since the insights remain the same, for brevity we do not extend the similar analysis to these
sections.
7.1.
Two Crops
We consider farmers that choose to plant one of two types of crops in each period. We use A
and B to denote the two crops. In line with the settings in the previous sections, we assume a
farmer’s production costs of planting crops A and B are c − δ and c − (1 − δ) respectively, where δ
is uniformly distributed on [0, 1]. In addition to capturing the heterogeneity of farmers’ production
costs, the assumed cost structure reflects two plausible facts: first, each farmer may have different
production costs for different crops; second, no farmer has cost advantages over other farmers with
both crops. The analysis can be extended to settings with production costs in more general forms.
In what follows we derive the quantity of crops A and B produced by naı̈ve and strategic farmers.
Under the assumed production cost structure, a naı̈ve farmer’s perceived utility of cultivating
B,n
crops A and B is uA,n
= pA
= pB
t
t−1 − (c − δ) and ut
t−1 − (c − (1 − δ)), respectively. Similarly, the
expected utility of planting crop A for a strategic farmer who rationally anticipates the market
B,s
price is uA,s
= pA
= pB
t
t − (c − δ), and the expected utility of planting crop B is ut
t − (c − (1 − δ)).
Compared with the single crop model in Section 4, the value of planting one crop here can be taken
27
as the endogenous outside option of planting the other.2 We assume uA,τ
and uB,τ
for τ ∈ {n, s}
t
t
are all positive to avoid degenerating into the single-crop model. A naı̈ve or strategic farmer will
cultivate crop A if uA,τ
≥ uB,τ
for τ ∈ {n, s}, and otherwise will cultivate crop B. Thus, in period
t
t
t, the quantities of crops A and B produced by naı̈ve farmers are
qtA,n = (1 − α)P r(uA,n
≥ uB,n
) and qtB,n = (1 − α)P r(uA,n
< uB,n
).
t
t
t
t
(19)
Similarly, the quantities of crops A and B produced by strategic farmers are
B,s
qtA,s = αP r(uA,s
≥ uB,s
= αP r(uA,s
< uB,s
t
t ) and qt
t
t ).
For ease of exposition, we assume crops A and B have the same market size Ω. The market clearing
prices of crops A and B are
A,n
B,n
pA
+ qtA,s and pB
+ qtB,s .
t = Ω − qt
t = Ω − qt
A
Proposition 14. Suppose |pB
0 − p0 | ≤ 1. The market prices of crops A and B in period t are
t
1
1
1−α
1
B
−
∆
(α)
and
p
=
Ω
−
+
∆
(α),
where
∆
(α)
=
−
pA
=
Ω
−
(P0B − P0A ). In particular:
t
t
t
t
t
2
2
2
1+α
(a) If there are no strategic farmers, i.e., α = 0, then the price in each market does not converge
but alternates between two prices. Specifically, for any n,
1 1 B
− P0 − P0A ,
2 2
1 1
A
B
p2n+1 = p2n = Ω − + P0B − P0A .
2 2
B
pA
2n = p2n+1 = Ω −
B
(b) If there exist strategic farmers, i.e., for all α > 0, then both market prices pA
t and pt converge
to Ω − 21 as t goes to ∞. Again, a tiny amount of strategic farmers stabilize the market price.
Proposition 14 shows that when farmers are able to choose one of two crops to plant in each
period, the market evolution is similar to the patterns in the single-crop models (see Propositions 1
and 3). When there are no strategic farmers, the market prices for both crops alternate between two
prices without convergence. When there are strategic farmers, the market price of each crop will
eventually converge to Ω − 12 , which is independent of the segment size of the strategic farmers, α.
Comparing with the single-crop models, the key difference is that now the farmers’ outside option
of cultivating one crop is endogenized as the value of cultivating the other crop. In particular, the
high price of one crop at the end of a period encourages naı̈ve farmers to plant this crop in the
2
There are papers that study the farmers’ planting decisions when there are crop rotation benefits, such as a larger
yield and lower farming costs on rotated farmland (see, e.g., Boyabatli et al. 2016 and related papers mentioned
therein). Our study focuses more on farmers’ planting decisions as reactions to the market-price dynamics.
28
next period, exacerbating the shortage of the other crop. Hence, the market prices for two crops
tend to wax and wane in alternation. In addition, the price fluctuation range of each crop is the
same as the original price difference between two crops.
The limiting price Ω −
1
2
in Proposition 14(b) implies that when there are strategic farmers,
eventually half of the farmers with cost advantage for crop A, i.e., δ > 12 , always cultivate crop A
and the other farmers always cultivate crop B in each period. Farmers are divided exactly in half,
because the market sizes of both crops are assumed to be the same as Ω. It can be shown that,
when the two crops have different market size, the market price of each crop will still converge as
long as: (i) there are strategic farmers; (ii) the difference between the market sizes of the two crops
is not too large. In the long run, the crop with a larger market size attracts more than half of the
farmers. In terms of the farmers’ welfare, the same statement as in Proposition 4 still holds in the
two-crops model, because strategic farmers continue to help stabilize the market prices for both
crops. That is, our finding about strategic farmers’ altruistic rationality continues to hold when
there are two crops.
7.2.
Random Yield
In this section, we extend the previous two-crop setting by considering yield uncertainty in farmers’
harvest. This uncertainty, which is attributed to uncontrollable external factors (e.g., weather,
water supply, pests), captures the fact that the quantity of crops a farmer harvests is often different
from the quantity actually planted. We denote the former as the random yield and the latter as
the production quantity. We capture the yield uncertainty by the ratio of the random yield to the
production quantity, and assume that the yield uncertainty in period t for all crops in the same
district is the same, denoted by γt .
In line with the setting in the previous subsection, we denote q̂tA,τ and q̂tB,τ for τ ∈ {n, s} as
B
the farmers’ production quantity of crop A and crop B for given market prices pA
t−1 and pt−1 ,
respectively. The random yields of two crops in period t are
QA,τ
= q̂tA,τ γt and QB,τ
= q̂tB,τ γt for τ ∈ {n, s} .
t
t
We assume that γt for all t ≥ 1 are independent and identically distributed random variables
with finite variance and that they are independent of farmers’ production costs. In the typical
production environment, γt ≤ 1, however, this is not a requirement of our model. We use γt = 1
to represent an average or normal external condition. The case of γt > 1 can capture a growing
season when crops grow very well, and γt < 1 captures a season with extreme weather or blight
that dooms a harvest. We assume E(γt ) = 1.
29
Now we derive the farmers’ production quantities and the corresponding market prices in a
period. For naı̈ve farmers, the production quantities q̂tA,n , q̂tB,n are exactly the same as qtA,n , qtB,n
defined in (19) under the assumption that E(γt ) = 1. In contrast, the production quantities of
strategic farmers depend on the rationally anticipated market prices, which are affected by the
yield. Specifically, the perceived expected utilities of planting crops A and B are defined as
uA,s
= E γPtA − (c − δ) and uB,s
= E γPtB − (c − (1 − δ)),
t
t
where γ is a representative random variable whose distribution is the same as γt for t ≥ 1 and is
independent of these γt , and PtA and PtB are the random market prices (see below). The corresponding production quantities of crops A and B are
q̂tA,s = αP r uA,s
≥ uB,s
and q̂tB,s = αP r uB,s
≥ uB,s
.
t
t
t
t
The resulting market clearing prices are now random, depending on the random yields of the two
crops, i.e.,
PtA = Ω − QA,s
− QA,n
and PtB = Ω − QB,s
− QB,n
.
t
t
t
t
The following proposition shows the convergence of the market prices.
(1−α)γ
2
Proposition 15. Suppose var[ln 1+αE[γ
2 ] ] < ∞ and (1 − 2α)E[γ ] ≤ 1, and define I(α, γ) :=
1−α
ln 1+αE[γ
2 ] + E[ln γ].
(a) If I(α, γ) ≤ 0, then the market prices PtA and PtB converge in probability towards the same
random variable P̄ := Ω − 21 γ, i.e.,
lim Pr(PtA − P̄ > ) = lim Pr(PtB − P̄ > ) = 0 for any > 0,
t→∞
t→∞
Moreover, the price difference between two crops converges to 0 as t goes to ∞.
(b) If I(α, γ) > 0, then the prices will not converge.
As (1 − 2α)E[γ 2 ] and I(α, γ) decrease in α > 0, Proposition 15 shows that compared with a market
in which all farmers are naı̈ve farmers, the market prices are more likely to converge when there
are also strategic farmers. When the convergence condition I(α, γ) ≤ 0 holds, the market prices
converge to a random variable P̄ , which is an affine transformation of the yield uncertainty γ. If
γ = 1, we recover the deterministic two-crop model, see Proposition 14(b). When the market prices
converge to the random variable P̄ := Ω − 12 γ, the price difference between two crops diminishes,
and hence the fluctuation of market prices remaining in the market comes purely from the yield
uncertainty.
30
8.
Conclusion
In developing countries, price volatility in agricultural markets is blocking the path to prosperity
for small-scale farmers. Although the burgeoning literature studies the value of disclosing market
information for price discovery and how that affects farmers’ welfare, little attention has been
paid to farmers’ heterogeneous capability in obtaining and processing the market information. In
reality, only farmers who are well-educated and not in remote areas may obtain and utilize the
market information in their decision making. It thus gives rise to the concern whether efforts to
disseminate market information treat all farmers fairly and whether the market is evolving in a
socially desirable way.
In this paper, we offer a stylized multi-period model to study farmers’ dynamic crop-planting
decisions and their impact on the market price. We consider both naı̈ve farmers who react shortsightedly to the latest market price, and strategic farmers who are able to collect and utilize
the information to anticipate rationally the near-future market price. Our model builds a microfoundation to capture the root cause of the price fluctuations observed in many agricultural markets. That is, because naı̈ve farmers base their crop planting decisions on the latest market price,
they cyclically change the crops they grow, thus leading to recurring over-production and price
slashing. We find that the forward-looking behavior of strategic farmers alleviates the herding effect
and is therefore both rational and altruistic in that it stabilizes the market price and increases all
farmers’ welfare. Surprisingly, even a small amount of strategic farmers can stabilize the market
and achieve the long-term benefits.
Our analysis also indicates the two-period, backward-looking (partially strategic) farmers may
be as effective in stabilizing the market price as forward-looking strategic farmers who have full
information on which to predict the near-future market price. The former farmers require less
sophisticated skills than the latter. In other words, some, but not necessarily full, rationality (or
information) could contribute to the market stabilization. However, in reality the market may not
be deterministic and stationary, and therefore backward-looking farmers need to use more historical
data to learn the market patterns. Moreover, if farmers have bounded rationality, their welfare can
be significantly reduced.
Another major finding is that even when it fails to cultivate strategic farmers, both social
entrepreneurs and for-profit firms can benefit all farmers as well as themselves by offering a preseason buyout contract to a fraction of farmers. Those farmers then only need to base their crop
decisions on the price offered by the firm. However, although a carefully designed contract benefits
all and leads to smaller income gaps among farmers, a poorly designed contract can distort the
31
market and hurt both the firm and farmers. We also extend our model to account for multiple
crops and yield uncertainty. We find that our results from the base model still hold qualitatively
in those extensions.
APPENDIX: Proofs
We provide proofs of the main results in this appendix and relegate the rest proofs to an online
appendix.
Proof of Proposition 1.
Recall that F (·) is the cumulative distribution function of uniform dis-
tribution on [0, 1], thus F (x) = x for any x ∈ [0, 1]. If 0 ≤ c − pt ≤ 1 for any t ≥ 0, then by (3),
we have pt = Ω − 1 + c − pt−1 = Ω − 1 + c − [Ω − 1 + c − pt−2 ] = pt−2 . Hence, p2i = p0 and p2i+1 =
Ω − 1 + c − p0 = 2p̄ − p0 . Therefore, pt = p̄ + (−1)t (p0 − p̄).
To complete the proof, we will show by induction that 0 ≤ c − pt ≤ 1 for any t ≥ 0. When t = 0,
0 ≤ c − pt ≤ 1 follows immediately from Assumption 2(b). For t ≥ 1, suppose for the inductive
hypothesis that 0 ≤ c − pt−1 ≤ 1, and we will show 0 ≤ c − pt ≤ 1. By (3) and Assumption 2(a),
we have Ω − 1 ≤ pt ≤ Ω for any t ≥ 0. By (3) and the hypothesis, we derive pt = Ω − 1 + c − pt−1 ,
thus c − pt = pt−1 − (Ω − 1). Given Ω − 1 ≤ pt−1 ≤ Ω, it follows that 0 ≤ c − pt ≤ 1. We have now
established that 0 ≤ c − pt ≤ 1 for any t ≥ 0. This completes the proof.
Proof of Proposition 2.
In each cycle i ≥ 1, the market prices are (p2i , p2i+1 ) = (p0 , 2p̄ − p0 ). Note
that p0 > 2p̄ − p0 because p0 > p̄. We divide the farmers based on their production cost endowment.
(a) Suppose c − δ ≤ 2p̄ − p0 . By (1), we have un2i = p2i−1 − (c − δ) = 2p̄ − p0 − (c − δ) ≥ 0 and un2i+1 =
p2i − (c − δ) = p0 − (c − δ) > 2p̄ − p0 − (c − δ) ≥ 0. Therefore, the farmer plants the crop in both
periods. Thus w(δ) = 21 [p2i − (c − δ)] + 21 [p2i+1 − (c − δ)] =
p2i +p2i+1
2
− (c − δ) = p̄ − (c − δ) ≥ p̄ − (2p̄ −
p0 ) = p0 − p̄ > 0. (b) Suppose 2p̄ − p0 < c − δ ≤ p0 . We have un2i = p2i−1 − (c − δ) = 2p̄ − p0 − (c − δ) < 0
and un2i+1 = p2i − (c − δ) = p0 − (c − δ) ≥ 0. Therefore, the farmer plants the crop only in period
2i + 1. Thus w(δ) = 12 [p2i+1 − (c − δ)] = 12 [2p̄ − p0 − (c − δ)] < 0. (c) Suppose p0 < c − δ. We have
un2i = p2i−1 − (c − δ) = 2p̄ − p0 − (c − δ) < p0 − (c − δ) < 0 and un2i+1 = p2i − (c − δ) = p0 − (c − δ) < 0.
Therefore, the farmer does not plant the crop in any period. Thus w(δ) = 0.
Proof of Proposition 3.
Suppose 0 ≤ c − pt ≤ 1 for any t ≥ 0. Then (9) reduces to pt = Ω − 1 +
α(c − pt ) + (1 − α)(c − pt−1 ). Through some algebra, pt =
Ω−1+c−(1−α)pt−1
1+α
= p̄ + (− 1−α
)t (p0 − p̄).
1+α
And it is clear that limt→∞ pt = p̄.
To complete the proof, we will show by induction that 0 ≤ c − pt ≤ 1 for any t ≥ 0. When t = 0,
0 ≤ c − p0 ≤ 1 follows immediately from Assumption 2(b). For t ≥ 1, suppose 0 ≤ c − pt−1 ≤ 1. By
(9), pt = Ω − 1 + αF (c − pt ) + (1 − α)(c − pt−1 ). By (9) and Assumption 2(a), we have Ω − 1 ≤ pt ≤ Ω
32
for any t ≥ 0 which will be used later. We now show 0 ≤ c − pt ≤ 1. Suppose c − pt < 0 and we
expect for a contradiction. Because c − pt < 0, pt = Ω − 1 + (1 − α)(c − pt−1 ), and thus c − pt =
−[Ω − 1 − αc − (1 − α)pt−1 ] ≥ 0 where the last inequality holds because Ω − 1 ≤ c (Assumption 1)
and Ω − 1 ≤ pt−1 , contradicting the supposition that c − pt < 0. Likewise, suppose c − pt > 1 and
again we expect for a contraction. Because c − pt > 1, pt = Ω − 1 + α + (1 − α)(c − pt−1 ), and thus
c − pt = −[Ω − α(c − 1) − (1 − α)pt−1 ] + 1 ≤ 1 where the last inequality holds because Ω ≥ c − 1
(Assumption 1) and Ω ≥ pt−1 , contradicting the supposition that c − pt > 1. Therefore, it must be
that 0 ≤ c − pt ≤ 1, which completes the proof.
Before we prove Proposition 4, we state two lemmas. For the proof of the lemmas, please see the
online appendix.
Lemma 1 (Long Run). Consider a farmer with production cost c − δ.
(a) If c − δ ≤ p̄, then the farmer plants the crop every period and w(δ) = p̄ − (c − δ) ≥ 0.
(b) If p̄ < c − δ, then the farmer does not plant the crop and w(δ) = 0.
Lemma 2 (Finite Horizon).
1. Consider a strategic farmer with production cost c − δ in
any cycle i ≥ 1. (a) If c − δ ≤ p2i+1 , then the farmer plants the crop in both periods and
w(δ) =
p2i +p2i+1
2
− (c − δ) > 0. (b) If p2i+1 < c − δ ≤ p2i , then the farmer plants the crop only in
period 2i and w(δ) = 21 [p2i − (c − δ)] > 0. (c) If p2i < c − δ, then the farmer does not plant the
crop in any period and w(δ) = 0.
2. Consider a naive farmer with production cost c − δ in any cycle i ≥ 1. (a) If c − δ ≤ p2i−1 , then
the farmer plants the crop in both periods and w(δ) =
p2i +p2i+1
2
− (c − δ) > 0. (b) If p2i−1 <
c − δ ≤ p2i , then the farmer plants the crop only in period 2i + 1 and w(δ) = 12 [p2i+1 − (c − δ)].
(c) If p2i < c − δ, then the farmer does not plant the crop in any period and w(δ) = 0.
Proof of Proposition 4.
We will discuss the welfare comparison from both the long run perspec-
tive and the short run (i.e., over a finite horizon) perspective, respectively.
Part 1(a): By Lemma 1, we see that in the long run, a strategic farmer earns the same welfare as
a naive farmer with the same production cost.
Part 1(b): We show that in the long run the naive farmer earns a higher welfare than that in the
absence of strategic farmers. Comparing the welfare stated in Lemma 1 with that in Proposition 2,
we find that: (a) for the naive farmers with 2p̄ − p0 < c − δ ≤ p̄, the welfare comparison is p̄ − (c −
δ) − 12 [2p̄ − p0 − (c − δ)] = 21 (p0 − p̄) + 21 [p̄ − (c − δ)] > 0, where the last inequality holds because
p0 > p̄ and c − δ ≤ p̄; (b) for the naive farmers with p̄ < c − δ ≤ p0 , the welfare comparison is
0 > 12 [2p̄ − p0 − (c − δ)]; (c) all the rest naive farmers obtain the same amount of welfare.
33
Part 2(a): We show that in the short run a strategic farmer earns a higher welfare than a naive
farmer with the same production cost. Note that p2i+1 > p2i−1 for any i ≥ 1. Comparing the welfare
stated in Part 1 with Part 2 in Lemma 2, we find that: (a) for the farmers with c − δ ≤ p2i−1 and
c − δ > p2i , the strategic farmer and the naive one obtain the same amount of welfare; (b) for the
farmers with p2i−1 < c − δ ≤ p2i+1 , the welfare comparison is
p2i +p2i+1
2
− (c − δ) > 21 [p2i+1 − (c − δ)]
because p2i > p2i+1 ≥ c − δ; (c) for the farmers with p2i+1 < c − δ ≤ p2i , the welfare comparison is
1
[p
2 2i
− (c − δ)] > 12 [p2i+1 − (c − δ)].
Part 2(b): We show that in the short run the naive farmer earns a higher welfare than that in
the absence of strategic farmers. Note that p2i−1 ≥ 2p̄ − p0 and p2i < p0 for any i ≥ 1. Comparing
the welfare stated in Lemma 2 with that in Proposition 2, we find that: (a) for the naive farmers
with c − δ ≤ 2p̄ − p0 , the welfare comparison is
p2i +p2i+1
2
− (c − δ) > p̄ − (c − δ) because p2i + p2i+1 >
2p̄ by (10); (b) for the naive farmers with 2p̄ − p0 < c − δ ≤ p2i−1 , the welfare comparison is
p2i +p2i+1
2
− (c − δ) > 0 > 12 [2p̄ − p0 − (c − δ)]; (c) for the farmers with p2i−1 < c − δ ≤ p2i , the welfare
comparison is 12 [p2i+1 − (c − δ)] > 21 [2p̄ − p0 − (c − δ)] because p2i+1 > p2i−1 ≥ 2p̄ − p0 ; (d) for the
farmers with p2i < c − δ ≤ p0 , the welfare comparison is 0 > 12 [2p̄ − p0 − (c − δ)]; (e) for the farmers
with p0 < c − δ , the welfare comparison is 0 = 0.
Combining Parts 1(a) and 2(a), we find that for every period, a strategic farmer earns a weakly
higher welfare than a naive farmer with the same production cost. Combining Parts 1(b) and 2(b),
we find that for every period, the naive farmer obtains a greater welfare than that in the absence
of strategic farmers. This completes the proof.
Before we prove Proposition 5, we state one lemma.
Lemma 3. In the model with backward looking strategic farmers, the recursive equation of market
price pt (t ≥ 1) is
pt = Ω − 1 + c − (1 −
Proof of Proposition 5.
α
α
)pt−1 − pt−2 .
2
2
(20)
We first show the existence of the limit. By Lemma 3,
pt − p̄ = −(1 −
α
α
)(pt−1 − p̄) − (pt−2 − p̄).
2
2
(21)
Thus |pt − p̄| ≤ (1 − α2 )|pt−1 − p̄| + α2 |pt−2 − p̄|, and so |pt − p̄| − |pt−1 − p̄| ≤ − α2 (|pt−1 − p̄| − |pt−2 − p̄|).
Therefore, limt→∞ |pt − p̄| exists. Remember that we expect to show limt→∞ pt − p̄ exists. Suppose
limt→∞ |pt − p̄| = A > 0 (Note that A 6= 0. Otherwise, we are done). Thus, for any given > 0, there
exists a T such that for any t > T ,
|pt − p̄| ∈ (A − , A + ).
(22)
34
Suppose for a contradiction that limt→∞ pt − p̄ does not exist. It follows from (22) that there must
exist a t1 with t1 − 1 > T and t1 − 2 > T such that pt1 −1 − p̄ ∈ (−A − , −A + ) and pt1 −2 − p̄ ∈
(A − , A + ). Therefore, by (21), pt1 − p̄ = −(1 − α2 )(pt1 −1 − p̄) − α2 (pt1 −2 − p̄) ∈ ((1 − α)A − 0 , (1 −
α)A + 0 ). Clearly, |pt1 − p̄| ∈
/ (A − , A + ), contradicting the fact (22). Consequently, limt→∞ pt − p̄
does exist.
Second, we compute the limit. Let limt→∞ pt − p̄ = B. Then by (21), we obtain B = 0.
Proof of Proposition 8.
We will show by induction that for i ≥ 1,
p2i = p̄ + (1 − α)i (p0 − p̄).
(23)
Since p0 > pb and 0 ≤ c − p0 ≤ 1, by (17), p1 = Ω − 1 + F (c − p0 ) = Ω − 1 + c − p0 . Thus c − 1 ≤ p1 ≤ c
because Ω − 1 ≤ p0 ≤ Ω (Assumption 2(a)). Moreover, p1 < p̄ = pb because p0 > p̄. In period 2,
type-S farmers accept the contract. Thus p2 = Ω − 1 + α(c − pb ) + (1 − α)(c − p1 ) = αpb + (1 − α)p0 =
p̄ + (1 − α)(p0 − p̄) > p̄ = pb where the last inequality holds because p0 > p̄. Moreover c − 1 ≤ p2 ≤ c
because c − 1 ≤ pb ≤ c (Assumption 4) and c − 1 ≤ p0 ≤ c (Assumption 2(b)). This establishes that
(23) holds when i = 1.
Suppose p2j = p̄ + (1 − α)j (p0 − p̄) = pb + (1 − α)j (p0 − pb ) and c − 1 ≤ p2j ≤ c. Clearly, p2j > pb .
Thus p2j+1 = Ω − 1 + c − p2j = pb − (p2j − pb ) < pb . Since Ω − 1 ≤ p2j ≤ Ω by (17), it follows that
c − 1 ≤ p2j+1 ≤ c. Therefore, p2(j+1) = Ω − 1 + c − αpb − (1 − α)p2j+1 = p̄ + (1 − α)j+1 (p0 − p̄).
Moreover, c − 1 ≤ p2(j+1) ≤ c because Ω − 1 ≤ pb = p̄ ≤ Ω and Ω − 1 ≤ p2j+1 ≤ Ω by (17).
i
We have now established that for any i ≥ 1, p2i = p̄ + (1 − α) (p0 − p̄). Note that p2i > pb and c −
1 ≤ p2i ≤ c, so p2i+1 = Ω − 1+c − p2i = p̄ − (p2i − p̄). One can easily check limt→∞ p2i = limt→∞ p2i+1 =
p̄.
Proof of Proposition 10.
Part 1: We will show that the SE could stabilize the market price
without a profit loss only if pb = p̄.
Suppose pb 6= p̄. By Proposition 7, we have the market price converges to either
Ω−1+c−αpb
2−α
or
Ω + α(c − 1 − pb ). Note that the price converges to Ω + α(c − 1 − pb ) only if pb > p̊. We have
Ω−1+c−αpb
2−α
− pb =
2(p̄−pb )
2−α
< 0 because pb > p̄, and Ω + α(c − 1 − pb ) − pb < Ω + α(c − 1 − p̊) − pb =
c − 1 − pb ≤ 0 because c − 1 ≤ pb (Assumption 4). Thus, the SE will suffer a profit loss in the long
run if pb 6= p̄.
Now we examine the case when pb = p̄. By Proposition 8, we have p2i+1 < p̄ = pb < p2i for any
i ≥ 1. Thus, type-S farmers accept the contract only in even periods, and the corresponding profit
b
b
b
gain in cycle i is (p2i − p̄)q2i
= (1 − α)i (p0 − p̄)q2i
≥ 0. One can check limi→∞ (p2i − p̄)q2i
= 0. Hence,
the SE obtains a positive profit over any finite horizon and the profit diminishes to 0 in the limit.
35
Part 2: We examine the welfare change.
Since the market price converges to p̄, which is the same as in the model with strategic farmers
in Section 4, both types of farmers obtain the same amount of welfare as shown in Lemma 1.
Therefore, by Proposition 4 we learn that both types of farmers receive a larger welfare than the
base model in the long run. Next we study the relation between incremental welfare and their cost
endowment. Comparing the welfare in Lemma 1 with that in Proposition 2, we obtain (a) for the
farmers with c − δ ≤ 2p̄ − p0 and c − δ > p0 , their welfare does not change; (b) for the farmers with
2p̄ − p0 < c − δ ≤ p̄, their incremental welfare is p̄ − (c − δ) − 21 [2p̄ − p0 − (c − δ)] = 12 [p0 − (c − δ)],
decreasing in their cost endowment; (c) for the farmers with p̄ < c − δ ≤ p0 , their incremental welfare
is 0 − 12 [2p̄ − p0 − (c − δ)] = 21 [p0 + (c − δ)] − p̄, increasing in their cost endowment. Therefore, the
incremental welfare is not monotone in farmer’s cost endowment c − δ.
Before we prove Proposition 11, we state one lemma.
Lemma 4 (Welfare Comparison). Suppose SE sets the price at pb = p̄. Consider a farmer
with production cost c − δ in each cycle i < ∞.
(a) If c − δ ≤ 2p̄ − p0 , then w̄in (δ) = w(δ) > w̄is (δ) > 0;
(b) If 2p̄ − p0 < c − δ ≤ p̄, then min{w̄is (δ), w̄in (δ)} > w(δ). In addition, for each δ, there exists
a threshold of cycle index i(δ) := min{i|p2i−1 ≥ c − δ } such that w̄in (δ) < w̄is (δ) for i < i(δ) and
w̄in (δ) ≥ w̄is (δ) otherwise;
(c) If p̄ < c − δ ≤ p0 , then w̄is (δ) = w̄in (δ) ≥ w(δ);
(d) If p0 < c − δ, then w̄is (δ) = w̄in (δ) = w(δ) = 0.
Proof of Proposition 11.
We first study the case when 2p̄ − p0 < c − δ2 ≤ p̄ and i ≥ i(δ). Checking
the proof of Lemma 4, we obtain that w̄is (δ2 ) =
addition, w̄is (δ1 ) =
pb +p2i+1
2
pb +p2i+1
2
− (c − δ2 ) and w̄in (δ2 ) = p̄ − (c − δ2 ). In
− (c − δ1 ) and w̄in (δ1 ) = p̄ − (c − δ1 ). Thus, w̄is (δ1 ) − w̄is (δ2 ) = δ1 − δ2 =
w̄in (δ1 ) − w̄in (δ2 ).
For all the other cases, just compare the welfare in Part (a) of Lemma 4 with that in Parts (b),
(c), and (d) respectively.
Proof of Proposition 12.
sition 3.
Compare the market dynamics in Proposition 8 with that in Propo-
Before we prove Proposition 15, we first derive some expressions and state one lemma.
The random yield quantity of two crops in period t are
QA,s
= γt q̂tA,s = γt αP r(uA,s
≥ uB,s
t
t
t ) = γt α[1 − F (
E[γt PtB ] − E[γt PtA ] + 1
)]
2
36
E[γt PtB ] − E[γt PtA ] + 1
)
2
A
P B − Pt−1
+1
)]
QA,n
= γt q̂tA,n = γt (1 − α)P r(uA,n
≥ uB,n
) = γt (1 − α)[1 − F ( t−1
t
t
t
2
A
P B − Pt−1
+1
QB,n
).
= γt q̂tB,n = γt (1 − α)P r(uA,n
< uB,n
) = γt (1 − α)F ( t−1
t
t
t
2
QB,s
= γt q̂tB,s = γt αP r(uA,s
< uB,s
t
t
t ) = γt αF (
The market clearing prices of Crops A and B are
A
P B − Pt−1
+1
E[γt PtB ] − E[γt PtA ] + 1
)] − (1 − α)γt [1 − F ( t−1
)]
2
2
(24)
B
A
B
A
P − Pt−1 + 1
E[γt Pt ] − E[γt Pt ] + 1
PtB = Ω − QB,s
− QB,n
= Ω − αγt F (
) − (1 − α)γt F ( t−1
). (25)
t
t
2
2
PtA = Ω − QA,s
− QA,n
= Ω − αγt [1 − F (
t
t
Thus,
A
+1
P B − Pt−1
E[γt PtB ] − E[γt PtA ] + 1
)] + (1 − α)γt [1 − 2F ( t−1
)]
(26)
2
2
A
P B − Pt−1
+1 2
E[γt PtB ] − E[γt PtA ] + 1
)] + (1 − α)[1 − 2F ( t−1
)] γt
PtB γt − PtA γt = α[1 − 2F (
2
2
A
P B − Pt−1
+1 E[γt PtB ] − E[γt PtA ] + 1
E[PtB γt ] − E[PtA γt ] = α[1 − 2F (
)] + (1 − α)[1 − 2F ( t−1
)] E[γt2 ].
2
2
(27)
PtB − PtA = αγt [1 − 2F (
Lemma 5. If (1 − 2α)E[γt2 ] ≤ 1, then

A

)
− (1−α)γt (P B − Pt−1

 1+αE(γt2 ) t−1
(1−α)γt
− 1+αE(γ
PtB − PtA =
2
t)


 (1−α)γ2t
B
A
if |Pt−1
− Pt−1
| ≤ 1,
B
A
if Pt−1
− Pt−1
> 1,
if
1+αE(γt )
Part of Proof of Proposition 15.
B
Pt−1
A
− Pt−1
(28)
< −1.
To save pages, we just show if I(α, γ) < 0, then the market
prices converge in probability towards P̄ . Observing (24) and (25), in order to see whether PtA and
PtB converge, it is essential to study whether PtB − PtA converges.
Part 1: We will prove the convergence of PtB − PtA if I(α, γ) < 0. By (28), we have
(1 − α)γt
(1 − α)γt−1
(1 − α)γ1
·
...
|P B − P0A |
2
2
1 + αE[γt ] 1 + αE[γt−1 ]
1 + αE[γ12 ] 0
t
X
(1 − α)γi
ln |PtB − PtA | ≤
ln
+ ln |P0B − P0A |.
2
1
+
αE[γ
]
i
i=1
|PtB − PtA | ≤
Doing the same operation to both sides of the above inequality, we obtain
(1−α)γ
ln |PtB − PtA | − ln |P0B − P0A | − tE[ln 1+αE[γ
2] ]
q
≤
(1−α)γ
t · var[ln 1+αE[γ
2] ]
Pt
(1−α)γi
i=1 [ln 1+αE[γ 2 ]
i
q
(1−α)γi
− E[ln 1+αE[γ
2 ] ]]
t · var[ln
i
(1−α)γ
]
1+αE[γ 2 ]
.
37
Note that the right hand side of the above inequality converges to the standard normal distribution
N (0, 1) as t → ∞, according to Central Limit Theorem. Thus,
lim P r(|PtB − PtA | ≥ ) = lim P r(ln |PtB − PtA | ≥ ln )
t→∞
= lim P r(
t→∞
ln |PtB
− PtA | − ln |P0B
q
t→∞
Pt
≤ lim P r(
t→∞
(1−α)γ
t · var[ln 1+αE[γ
2] ]
(1−α)γi
i=1 [ln 1+αE[γ 2 ]
i
q
(1−α)γ
− P0A | − tE[ln 1+αE[γ
2] ]
(1−α)γi
− E[ln 1+αE[γ
2 ] ]]
i
(1−α)γ
t · var[ln 1+αE[γ
2] ]
≥
≥
(1−α)γ
ln − ln |P0B − P0A | − tE[ln 1+αE[γ
2] ]
q
)
(1−α)γ
t · var[ln 1+αE[γ
]
2]
(1−α)γ
ln − ln |P0B − P0A | − tE[ln 1+αE[γ
2] ]
q
)
(1−α)γ
]
t · var[ln 1+αE[γ
2]
=P r(N (0, 1) ≥ ∞) = 0,
where the second last equality holds because the left hand side above is a standard normal distri(1−α)γ
bution and in the right hand side E[ln 1+αE[γ
2 ] ] = I(α, γ) < 0.
Part 2: We will prove the convergence of PtB and PtA if I(α, γ) < 0.
By Part 1, we have if I(α, γ) < 0, then the process PtB − PtA will converge to 0 in probability. It
follows immediately by (26) that limt→∞ P r(|E[PtB γt ] − E[PtA γt ]| < ) = 1. Plugging back into (24)
and (25) yields limt→∞ P r(|PtA − P̄ | > ) = limt→∞ P r(|PtB − P̄ | > ) = 0.
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1
Online Appendix
Altruistic Rationality: The Value of Strategic Farmers,
Social Entrepreneurs and For-Profit Firms in Crop Planting Decisions
OA.1.
Proofs
Proof of Lemma 1.
In the long run, the market price is stabilized at p̄ by Proposition 3.
If c − δ ≤ p̄, then unt = ust = p̄ − (c − δ) ≥ 0, thus both types of farmers plant the crop every period
and w(δ) = p̄ − (c − δ) ≥ 0.
If p̄ < c − δ, then unt = ust = p̄ − (c − δ) < 0, thus both types of farmers do not plant the crop in
any period and w(δ) = 0.
Proof of Lemma 2.
In each cycle i, the market prices are
1 − α 2i
) (p0 − p̄)
1+α
1 − α 2i+1
)
(p0 − p̄).
p2i+1 = p̄ + (−
1+α
p2i = p̄ + (−
Note that
p2i > p̄ > p2i+1
(OA.1)
p2i + p2i+1 > 2p̄.
(OA.2)
Part 1: We consider the strategic farmer.
If c − δ ≤ p2i+1 , then
us2i = p2i − (c − δ) > p2i+1 − (c − δ) ≥ 0
us2i+1 = p2i+1 − (c − δ) ≥ 0
1
p2i + p2i+1
1
− (c − δ) > p̄ − (c − δ) > 0.
w(δ) = [p2i − (c − δ)] + [p2i+1 − (c − δ)] =
2
2
2
The second last inequality above holds because of (OA.2).
If p2i+1 < c − δ ≤ p2i , then
us2i = p2i − (c − δ) ≥ 0
us2i+1 = p2i+1 − (c − δ) < 0
1
w(δ) = [p2i − (c − δ)] ≥ 0.
2
If p2i < c − δ, then
us2i = p2i − (c − δ) < 0
2
us2i+1 = p2i+1 − (c − δ) < p2i − (c − δ) < 0
w(δ) = 0.
Part 2: We consider the naive farmer.
If c − δ ≤ p2i−1 , then
un2i = p2i−1 − (c − δ) ≥ 0
un2i+1 = p2i − (c − δ) > p2i−1 − (c − δ) ≥ 0
1
1
p2i + p2i+1
w(δ) = [p2i − (c − δ)] + [p2i+1 − (c − δ)] =
− (c − δ) > p̄ − (c − δ) > 0.
2
2
2
If p2i−1 < c − δ ≤ p2i , then
un2i = p2i−1 − (c − δ) < 0
un2i+1 = p2i − (c − δ) > 0
1
w(δ) = [p2i+1 − (c − δ)].
2
If p2i < c − δ, then
un2i = p2i−1 − (c − δ) < p2i − (c − δ) < 0
un2i+1 = p2i − (c − δ) < 0
w(δ) = 0.
This completes the proof.
Proof of Lemma 3.
If 0 ≤ c − pt ≤ 1 and 0 ≤ c −
pt = Ω − 1 + (1 − α)(c − pt−1 ) + α(c −
pt +pt−1
2
≤ 1 for any t ≥ 0, then (12) reduces to
pt−1 + pt−2
α
α
) = Ω − 1 + c − (1 − )pt−1 − pt−2 .
2
2
2
To complete the proof, we will show by induction that 0 ≤ c − pt ≤ 1 and 0 ≤ c −
pt +pt−1
2
≤ 1 for
any t ≥ 0.
By Assumption 2(b) and the convention that pi = p0 for any i < 0, we have 0 ≤ c − p0 ≤ 1 and
0≤c−
p0 +p−1
2
≤ 1.
By (12) and Assumption 2(a), we have
Ω − 1 ≤ pt ≤ Ω
for any t ≥ 0.
For any given t ≥ 1, suppose 0 ≤ c − pt−1 ≤ 1 and 0 ≤ c −
pt = Ω − 1 + c − (1 − α)pt−1 − α
pt−1 +pt−2
2
≤ 1, then by (12),
pt−2 + pt−1
.
2
3
Hence,
c − pt =
p
+p
(1 − α)pt−1 + α t−1 2 t−2 − (Ω − 1) ≥ 0 by pt ≥ Ω − 1 for any t
p
+p
(1 − α)pt−1 + α t−1 2 t−2 − Ω + 1 ≤ 1 by pt ≤ Ω for any t
(OA.3)
So
0≤c−
pt + pt−1
≤ 1.
2
We have now established that 0 ≤ c − pt ≤ 1 and 0 ≤ c −
the proof.
pt +pt−1
2
≤ 1 for any t ≥ 0. This completes
Before we prove Proposition 6, we state one lemma.
Lemma OA.1. The recursive equation of market price pt (t ≥ 1) is
1 + e(pt −c+1)/β
= Ω − (1 − α)(1 − c + pt−1 ),
1 + e(pt −c)/β
pt + αβ ln
(OA.4)
if either of the following conditions holds,
(Ω−c)/β
1+e
(a) if c ≤ Ω < c + 1 and β ln 1+e
(Ω−1−c)/β ≥ Ω − c.
(Ω−c+1)/β
≤ Ω + 1 − c.
(b) if c − 1 < Ω < c and β ln 1+e
1+e(Ω−c)/β
Proof of Lemma OA.1.
We consider two cases.
(Ω−c)/β
1+e
Case 1: Suppose c ≤ Ω < c + 1 and β ln 1+e
(Ω−1−c)/β ≥ Ω − c.
We will establish that 0 ≤ c − pt ≤ 1 for any t ≥ 0, from which (OA.4) follows immediately by
(13).
By (13), we have
Ω − 1 ≤ pt ≤ Ω
for any t ≥ 0.
(OA.5)
It follows that
c − pt ≤ c − (Ω − 1) ≤ 1
for any t ≥ 0,
(OA.6)
where the last inequality holds because c ≤ Ω. Hence, we just need to show c − pt ≥ 0 for any t ≥ 0.
When t = 0, c − p0 ≥ 0 follows immediately from Assumption 2(b).
(Ω−c)/β
1+e
Before we continue to proceed the induction, we first show if β ln 1+e
(Ω−1−c)/β ≥ Ω − c, then
β ln
1 + e(pt −c+1)/β
≥ Ω − c,
1 + e(pt −c)/β
for any pt
(x−c+1)/β
(OA.7)
which will be used to show c − pt ≥ 0 later. Let g(x) = β ln 1+e
. It is easy to verify g 0 (x) > 0.
1+e(x−c)/β
Hence (OA.7) holds because pt ≥ Ω − 1.
4
For any given t ≥ 1, suppose c − pt−1 ≥ 0, then by (13),
pt = Ω − (1 − α)(1 − c + pt−1 ) − αβ ln
1 + e(pt −c+1)/β
.
1 + e(pt −c)/β
Thus,
c − pt = (1 − α)(pt−1 − Ω + 1) + α(c − Ω + β ln
1 + e(pt −c+1)/β
) ≥ 0.
1 + e(pt −c)/β
The above inequality holds because pt−1 ≥ Ω − 1 and (OA.7).
We have now shown that 0 ≤ c − pt ≤ 1 for any t ≥ 0. Consequently, (OA.4) holds.
(Ω−c+1)/β
Case 2: Suppose c − 1 < Ω < c and β ln 1+e
≤ Ω + 1 − c.
1+e(Ω−c)/β
Again, we will establish that 0 ≤ c − pt ≤ 1 for any t ≥ 0, from which (OA.4) follows.
We have
c − pt ≥ c − Ω > 0.
for any t ≥ 0
(OA.8)
where the first inequality holds because of (OA.5) and the second inequality holds because c > Ω.
Hence, we just need to show c − pt ≤ 1 for any t ≥ 0.
When t = 0, c − p0 ≤ 1 follows immediately from Assumption 2(b).
(Ω−c+1)/β
Before we continue to proceed the induction, we first show if β ln 1+e
≤ Ω + 1 − c, then
1+e(Ω−c)/β
β ln
1 + e(pt −c+1)/β
≤ Ω + 1 − c,
1 + e(pt −c)/β
(OA.9)
(x−c+1)/β
which will be used to show c − pt ≤ 1 later. Let g(x) = β ln 1+e
. Since g 0 (x) > 0, (OA.9)
1+e(x−c)/β
holds because pt ≤ Ω.
For any given t ≥ 1, suppose c − pt−1 ≤ 1, then by (13),
pt = Ω − (1 − α)(1 − c + pt−1 ) − αβ ln
1 + e(pt −c+1)/β
.
1 + e(pt −c)/β
Thus
c − pt = (1 − α)(pt−1 − Ω) + α(c − Ω − 1 + β ln
1 + e(pt −c+1)/β
) + 1 ≤ 1.
1 + e(pt −c)/β
The above inequality holds because pt−1 ≤ Ω and (OA.9).
We have now shown that 0 ≤ c − pt ≤ 1 for any t ≥ 0. Consequently, (OA.4) holds.
Proof of Proposition 6.
First we show if c − 12 ≤ Ω ≤ c + 12 , then (OA.4) holds. To this end, it
suffices to show if c − 12 ≤ Ω ≤ c + 12 , either of the conditions in Lemma OA.1 holds. We consider
two cases separately.
5
(Ω−c)/β
1+e
Case 1: Suppose c ≤ Ω ≤ c + 21 . We want to show β ln 1+e
(Ω−1−c)/β ≥ Ω − c.
We now take a brief detour and observe that
1 + e(Ω−c)/β
1
− 1 = (Ω−c)/β
1 + e(Ω−c)/β − e(Ω−c)/β [1 + e(Ω−c−1)/β ]
(Ω−c)/β
(Ω−c−1)/β
(Ω−c−1)/β
e
[1 + e
]
e
[1 + e
]
2(Ω−c)−1 1
= (Ω−c)/β
1−e β
≥0
(OA.10)
e
[1 + e(Ω−c−1)/β ]
where the last inequality holds because Ω ≤ c + 12 . Hence, it follows that
1 + e(Ω−c)/β
1 + e(Ω−c)/β
(Ω−c)/β
=
β
ln[e
]
+
β
ln
− β ln[e(Ω−c)/β ]
1 + e(Ω−1−c)/β
1 + e(Ω−1−c)/β
1 + e(Ω−c)/β
=Ω − c + β ln (Ω−c)/β
≥Ω−c
e
[1 + e(Ω−c−1)/β ]
β ln
where the last inequality holds because of (OA.10).
Therefore, if c ≤ Ω ≤ c + 12 , then the first condition in Lemma OA.1 holds, and thus (OA.4) holds.
(Ω−c+1)/β
Case 2: Suppose c − 12 ≤ Ω ≤ c. We want to show β ln 1+e
≤ Ω − c + 1.
1+e(Ω−c)/β
We have
1
1 + e(Ω−c+1)/β
1 + e(Ω−c+1)/β − e(Ω−c+1)/β [1 + e(Ω−c)/β ]
− 1 = (Ω−c+1)/β
(Ω−c+1)/β
(Ω−c)/β
(Ω−c)/β
e
[1 + e
]
e
[1 + e
]
2(Ω−c)+1 1
= (Ω−c+1)/β
1−e β
≤0
(OA.11)
e
[1 + e(Ω−c)/β ]
where the last inequality holds because Ω ≥ c − 21 . Hence, it follows that
1 + e(Ω−c+1)/β
1 + e(Ω−c+1)/β
(Ω−c+1)/β
=
β
ln[e
]
+
β
ln
− β ln[e(Ω−c+1)/β ]
1 + e(Ω−c)/β
1 + e(Ω−c)/β
1 + e(Ω−c+1)/β
=Ω − c + 1 + β ln (Ω−c+1)/β
≤Ω−c+1
e
[1 + e(Ω−c)/β ]
β ln
where the last inequality holds because of (OA.11).
Therefore, if c − 12 ≤ Ω ≤ c, then the second condition in Lemma OA.1 holds, and thus (OA.4)
holds. We have now established that (OA.4) holds when c − 12 ≤ Ω ≤ c + 12 .
Second, we show the convergence of pt . By (OA.4), we have
1 + e(pt −c+1)/β
= Ω − (1 − α)(1 − c + pt−1 )
1 + e(pt −c)/β
1 + e(pt−1 −c+1)/β
pt−1 + αβ ln
= Ω − (1 − α)(1 − c + pt−2 ).
1 + e(pt−1 −c)/β
pt + αβ ln
Thus
pt − pt−1 + αβ(ln
1 + e(pt −c+1)/β
1 + e(pt−1 −c+1)/β
−
ln
) = −(1 − α)(pt−1 − pt−2 ).
1 + e(pt −c)/β
1 + e(pt−1 −c)/β
(OA.12)
6
(x−c+1)/β
. Taking derivative yields
Let f (x) = ln 1+e
1+e(x−c)/β
f 0 (x) =
1
e(x−c+1)/β − e(x−c)/β
.
β [1 + e(x−c+1)/β ][1 + e(x−c)/β ]
By Mean Value Theorem, we have
ln
1 + e(pt −c+1)/β
1 + e(pt−1 −c+1)/β
−
ln
= f (pt ) − f (pt−1 ) = f 0 (ξt )(pt − pt−1 ),
1 + e(pt −c)/β
1 + e(pt−1 −c)/β
(OA.13)
where min{pt , pt−1 } ≤ ξt ≤ max{pt , pt−1 }. Plugging (OA.13) back into (OA.12) yields
(pt − pt−1 )[1 + αβf 0 (ξt )] = −(1 − α)(pt−1 − pt−2 )
1−α
pt − pt−1 = −
(pt−1 − pt−2 ).
1 + αβf 0 (ξt )
Since f 0 (x) > 0 for any x, 1 + αβf 0 (ξt ) > 1 and thus
1−α
1+αβf 0 (ξt )
< 1 − α < 1. Therefore, limt→∞ pt
does exist.
Let limt→∞ pt = X. Then by (OA.4), we obtain the following equation
X + αβ ln
1 + e(X−c+1)/β
= Ω − (1 − α)(1 − c + X).
1 + e(X−c)/β
Observe that f (X) is increasing in X, so the left hand side of the above equation is increasing in
X. Given the right hand side is decreasing in X, there is a unique solution to the above equation. Lemma OA.2. If ϕt = 21 , then
pt = Ω − (1 − α)(1 − c) −
Proof of Lemma OA.2.
Similar to the proof of Lemma 3.
Lemma OA.3. If β = ∞, then limt→∞ pt =
with production cost c − δ. (i) If c − δ ≤
and obtains a welfare of
α
− (1 − α)pt−1 .
2
Ω−(1−α)(1−c)−α/2
.
2−α
Ω−(1−α)(1−c)−α/2
,
2−α
Ω−(1−α)(1−c)−α/2
2−α
(OA.14)
In this case, consider a naive farmer
then the farmer plants the crop each period
− (c − δ). (ii) If c − δ >
Ω−(1−α)(1−c)−α/2
,
2−α
then the farmer
does not plant the crop and obtains a welfare of 0. The strategic farmer with production cost c − δ
− (c − δ)].
obtains a welfare of 21 [ Ω−(1−α)(1−c)−α/2
2−α
Proof of Lemma OA.3.
Part 1: We will show the convergence if β = ∞.
If β = ∞, then strategic farmers randomly decide whether to plant or not. That is, ϕt =
any t. Hence, by Lemma OA.2,
α
− (1 − α)pt−1
2
α
pt−1 = Ω − (1 − α)(1 − c) − − (1 − α)pt−2 .
2
pt = Ω − (1 − α)(1 − c) −
1
2
for
7
Thus
pt − pt−1 = −(1 − α)(pt−1 − pt−2 ).
It is apparent that limt→∞ pt does exist.
Let limt→∞ pt = X. By (OA.14), we obtain X =
Ω−(1−α)(1−c)− α
2
2−α
.
Part 2: We examine farmers’ welfare.
Since the market price is stabilized at
Ω−(1−α)(1−c)−α/2
,
2−α
it is easy to see that a naive farmer
will plant if his production cost is lower than the market price and will not plant otherwise.
Since the strategic farmer randomly decides whether to plant or not, he obtains a welfare of
1 Ω−(1−α)(1−c)−α/2
[
2
2−α
− (c − δ)].
Lemma OA.4. (a) If β = ∞, some naive farmers and strategic farmers may obtain a lower
welfare than naive farmer in the base model.
(b) If Ω ≤ c, both naive and strategic farmers when β = ∞ receive a lower welfare than that with
strategic farmers being fully rational (i.e., β = 0).
Proof of Lemma OA.4.
Part (a): By Proposition 2, we have that in the base model, a naive
farmer with c − δ ≤ 2p̄ − p0 obtains a welfare of p̄ − (c − δ). By Lemma OA.3, we have that when
β = ∞, a naive farmer with c − δ ≤
Since
Ω−(1−α)(1−c)− α
2
2−α
Ω−(1−α)(1−c)− α
2
2−α
Ω−(1−α)(1−c)− α
2
− (c − δ).
2−α
Ω−(1−α)(1−c)− α
2
min{2p̄ − p0 ,
}
2−α
obtains a welfare of
< p̄, it is fair to say the naive farmers with c − δ ≤
obtains a lower welfare than that with all farmers being naive.
By Proposition 2, we have that in the base model, a naive farmer with c − δ > p0 obtains
a welfare of 0. By Lemma OA.3, we have that when β = ∞, the strategic farmer obtains
1 Ω−(1−α)(1−c)−α/2
[
− (c
2
2−α
Ω−(1−α)(1−c)− α
2
max{p0 ,
}, they obtain
2−α
a welfare of
− δ)]. Therefore, for the strategic farmers with c − δ >
a lower welfare than the naive farmers in the base model.
Part (b): Comparing the limiting price when β = ∞ with that when β = 0, we obtain
Ω − (1 − α)(1 − c) − α2 Ω − 1 + c α(Ω − c)
−
=
≤ 0.
2−α
2
2(2 − α)
If β = ∞, the naive farmers face a lower limiting market price than that in the model with
fully rational strategic farmers (β = 0), thus each of them obtains a lower welfare. Meanwhile, the
strategic farmers randomly decide to cultivate or not (with probability 1/2) and obtains a welfare
α
1 Ω−(1−α)(1−c)− 2
[
− (c − δ)].
2
2−α
Ω−(1−α)(1−c)− α
2
can check 12 [
2−α
Comparing with the strategic farmers’ welfare shown in Lemma 1, one
− (c − δ)] ≤ p̄ − (c − δ) if c − δ ≤ p̄ and 12 [
Ω−(1−α)(1−c)− α
2
2−α
− (c − δ)] ≤ 0
otherwise. Therefore, each strategic farmer when β = ∞ obtains a lower welfare as well. 8
Before we prove Proposition 7, we state two lemmas and their proofs.
Lemma OA.5. If c ≤ Ω < c + 1 and pb ≥ p̄, then
lim pt =
t→∞
Proof of Lemma OA.5.
Ω − 1 + c − αpb
.
2−α
Observe that
c − Ω ≤ 0,
0 < c − (Ω − 1) ≤ 1.
(OA.15)
By (17) and Assumption 2(a), we have pt ≥ Ω − 1 for any t ≥ 0. Thus
c − pt ≤ c − (Ω − 1) ≤ 1
for any t ≥ 0.
(OA.16)
We consider two cases.
Case 1: Suppose p0 < pb .
Note that p0 < pb ≤ c where pb ≤ c holds because of Assumption 4. Since pb > p0 , in period 1
type-S farmers accept the contract. Thus by (17),
p1 = Ω − 1 + c − αpb − (1 − α)p0 .
(OA.17)
Subcase 1: Suppose p1 < pb .
We will show by induction that
pt =
1 − [−(1 − α)]t
[Ω − 1 + c − αpb ] + [−(1 − α)]t p0
2−α
for any t ≥ 1.
(OA.18)
Note that pb ≤ c, so it follows by the supposition that p1 < c. Together with (OA.16), we have
that 0 ≤ c − p1 ≤ 1. (OA.17) indicates that (OA.18) holds when t = 1.
For any t ≥ 2, suppose pt−1 =
1−[−(1−α)]t−1
[Ω − 1 + c − αpb ] + [−(1 − α)]t−1 p0 ,
2−α
0 ≤ c − pt−1 ≤ 1 and
pt−1 < pb . Thus
1 − [−(1 − α)]t
[Ω − 1 + c − αpb ] + [−(1 − α)]t p0
2−α
1 − [−(1 − α)]t b
1 − [−(1 − α)]t
1 − [−(1 − α)]t b
<
[p + (1 − α)p0 ] + [−(1 − α)]t p0 =
p + (1 −
)p0
2−α
2−α
2−α
< pb ≤ c
pt = Ω − 1 + α(c − pb ) + (1 − α)(c − pt−1 ) =
where the first inequality holds because p1 = Ω − 1 + c − αpb − (1 − α)p0 < pb and the second
inequality holds because p0 < pb .
We have now proved that for any t ≥ 1,
pt =
1 − [−(1 − α)]t
[Ω − 1 + c − αpb ] + [−(1 − α)]t p0 .
2−α
9
It can be checked that limt→∞ pt =
Ω−1+c−αpb
.
2−α
Subcase 2: Suppose p1 ≥ pb .
If p1 = p̄, then it must be that p1 = pb and pb = p̄. In this case, we have pt = p̄ for any t ≥ 1. That
is, the price converges to
Ω−1+c
2
=
Ω−1+c−αpb
.
2−α
Next we study the case if p1 > p̄. Our objective is to
show there exists a i1 ≥ 1 such that p2i1 < pb and p2i1 +1 < pb , then the analysis follows the same as
Subcase 1.
Note that p0 < pb and p0 ≥ Ω − 1, so pb ≥ Ω − 1. Plus p0 ≥ Ω − 1, it follows by (OA.17) that p1 ≤ c.
Together with (OA.16), we have that 0 ≤ c − p1 ≤ 1.
Since p1 ≥ pb , no type-S farmers accept the contract. Thus
p2 = Ω − 1 + F (c − p1 ) = Ω − 1 + c − p1 < p̄ ≤ pb .
Moreover, 0 ≤ c − p2 ≤ 1 because p2 < pb ≤ c and (OA.16).
In period 3, type-S farmers accept the contract and thus
p3 = Ω − 1 + c − αpb − (1 − α)p2 = α(Ω − 1 + c − pb ) + (1 − α)p1 ≤ c
where the last inequality holds because pb > p0 ≥ Ω − 1 and p1 ≤ c. Hence, with (OA.16), we have
0 ≤ c − p3 ≤ 1.
If p3 < pb , then we are done. Suppose p3 ≥ pb , then
p4 = Ω − 1 + F (c − p3 ) = Ω − 1 + c − p3 = (1 − α)(Ω − 1 + c − p1 ) + αpb < pb
where the last inequality holds because p2 = Ω − 1 + c − p1 < pb . Moreover, 0 ≤ c − p4 ≤ 1 because
p4 < pb ≤ c and (OA.16).
In period 5, type-S farmers accept the contract and thus
p5 = Ω − 1 + c − αpb − (1 − α)p4 = [1 − (1 − α)2 ](Ω − 1 + c − pb ) + (1 − α)2 p1 < c
where the last inequality holds because pb > p0 ≥ Ω − 1 and p1 ≤ c.
Continuing in this fashion for i ≥ 1, we obtain that if p2i−1 ≥ pb and p2i−1 > p̄,
p2i = Ω − 1 + c − p2i−1 < pb
p2i+1 = [1 − (1 − α)i ](Ω − 1 + c − pb ) + (1 − α)i p1 .
If pb > p̄, then Ω − 1 + c − pb < pb , and we see that there must exist a i1 such that p2i1 < pb and
p2i1 +1 < pb . Then the analysis follows the Subcase 1 above. If pb = p̄, then Ω − 1 + c − pb = p̄. We
could see that limi→∞ p2i = limi→∞ p2i+1 = Ω − 1 + c − p̄ = p̄ =
Ω−1+c−αpb
.
2−α
10
If p2i−1 = pb = p̄, then Ω − 1 + c − pb = p̄. Thus pt = Ω − 1 + c − pt−1 = p̄ =
Ω−1+c−αpb
2−α
for any
t ≥ 2i − 1.
Case 2: Suppose p0 ≥ pb .
If p0 = p̄, then it must be that p0 = pb and pb = p̄. In this case, we have pt = p̄ for any t ≥ 1. That
is, the price converges to
Ω−1+c
2
=
Ω−1+c−αpb
.
2−α
Hence we just need to study if p0 > p̄.
Since p0 ≥ pb , no farmers accept the contract. Thus
p1 = Ω − 1 + c − p0 < p̄ ≤ pb .
If p2 ≥ pb , then the analysis follows the Subcase 2 of Case 1. If p2 < pb , then the analysis follows
the Subcase 1 of Case 1.
This completes the proof.
Lemma OA.6. If c − 1 < Ω < c and p̄ ≤ pb ≤ p̊, then
lim pt =
t→∞
Proof of Lemma OA.6
Ω − 1 + c − αpb
.
2−α
Observe that
0 < c − Ω < 1,
c − (Ω − 1) > 1.
(OA.19)
By (17) and Assumption 2(a), we have pt ≤ Ω for any t ≥ 0. Thus
c − pt ≥ c − Ω > 0
Since pb ≤ p̊ =
Ω−(1−α)(c−1)
,
α
for any t ≥ 0.
(OA.20)
we have
Ω − αpb ≥ (1 − α)(c − 1).
(OA.21)
We consider two Cases.
Case 1: Suppose p0 < pb .
We have
p1 = Ω − 1 + c − αpb − (1 − α)p0 .
Subcase 1: Suppose p1 < pb .
Subsubcase1: Suppose Ω − αpb − (1 − α)p0 ≥ 0.
We will show by induction that pt =
1−[−(1−α)]t
(Ω − 1 + c − αpb ) + [−(1 − α)]t p0
2−α
for any t ≥ 2.
11
By supposition, p1 ≥ c − 1. Thus we have 0 ≤ c − p1 ≤ 1 by (OA.20). Hence
p2 = Ω − 1 + α(c − pb ) + (1 − α)F (c − p1 ) = Ω − 1 + α(c − pb ) + (1 − α)(c − p1 )
= α(Ω − 1 + c − αpb ) + (1 − α)2 p0 ≥ α[c − 1 + (1 − α)p0 ] + (1 − α)2 p0 ≥ c − 1
where the first inequality holds because of our supposition and the second inequality holds because
p0 ≥ c − 1. And
p2 − pb = α(Ω − 1 + c − αpb ) + (1 − α)2 p0 − pb = 2αp̄ + (1 − α)2 p0 − (1 + α2 )pb < 0
where the last inequality holds because p̄ ≤ pb and p0 < pb . This established that the induction
equation holds when t = 2.
Suppose pt−1 =
1−[−(1−α)]t−1
(Ω
2−α
− 1 + c − αpb ) + [−(1 − α)]t−1 p0 , c − 1 ≤ pt−1 ≤ c, and pt−1 < pb .
Then
pt = Ω − 1 + c − αpb − (1 − α)pt−1 =
1 − [−(1 − α)]t
(Ω − 1 + c − αpb ) + [−(1 − α)]t p0
2−α
1 − [−(1 − α)]t
(c − 1 + (1 − α)p0 ) + [−(1 − α)]t p0
2−α
1 − [−(1 − α)]t
1 − [−(1 − α)]t
(c − 1) + (1 −
)p0
=
2−α
2−α
≥c−1
≥
where the first inequality holds because of our supposition and the second inequality holds because
p0 ≥ c − 1. And
1 − (α − 1)t
(Ω − 1 + c − αpb ) + [−(1 − α)]t p0 − pb
2−α
1 − [−(1 − α)]t
<
[(1 − α)p0 + pb ] + [−(1 − α)]t p0 − pb
2−α
1 − α + [−(1 − α)]t
=
(p0 − pb )
2−α
< 0.
pt − pb =
[by p1 < pb ]
[by p0 < pb ]
Hence, we have established that for any t ≥ 2,
pt =
1 − [−(1 − α)]t
(Ω − 1 + c − αpb ) + [−(1 − α)]t p0 .
2−α
It can be checked that limt→∞ pt =
Ω−1+c−αpb
.
2−α
Subsubcase 2: Suppose Ω − αpb − (1 − α)p0 ≤ 0.
By this supposition, p1 ≤ c − 1. Thus
p2 = Ω − 1 + α(c − pb ) + (1 − α)F (c − p1 ) = Ω − 1 + α(c − pb ) + 1 − α
12
= Ω − αpb + α(c − 1) < (1 − α)p0 + α(c − 1) ≤ p0 < pb .
where the first inequality holds because of our supposition. And
p2 − (c − 1) = Ω − αpb + α(c − 1) − (c − 1) = Ω − αpb − (1 − α)(c − 1) ≥ 0
where the last inequality holds because of (OA.21). Moreover,
Ω − αpb − (1 − α)p2 = α[Ω − αpb − (1 − α)(c − 1)] ≥ 0.
And
p3 − pb = Ω − 1 + c − αpb − (1 − α)p2 − pb = Ω − 1 + c − (1 − α)p2 − (1 + α)pb
= 2αp̄ + (1 − α)2 (c − 1) − (1 + α)2 pb < 0
where the last inequality holds because c − 1 < p̄ ≤ pb .
The preceding four inequalities imply that (p2 , p3 ) has the same property as (p0 , p1 ) in Subsubcase
1. Hence, the analysis follows the same.
Subcase 2: Suppose p1 ≥ pb .
If p1 = p̄, then it must be that p1 = pb and pb = p̄. In this case, we have pt = p̄ for any t ≥ 1. That
is, the price converges to
Ω−1+c
2
=
Ω−1+c−αpb
.
2−α
Next we study the case if p1 > p̄. Our objective is to
show that there exists a i2 ≥ 1 such that p2i2 < pb and p2i2 +1 < pb , then it follows the analysis in
Subcase 1.
Note that c − 1 <
Ω−1+c
2
= p̄ < p1 . Since p1 ≥ pb , in period 2 no type-S farmers accept the contract,
thus
p2 = Ω − 1 + F (c − p1 ) = Ω − 1 + c − p1 .
Moreover, p2 <
Ω−1+c
2
≤ pb because p1 >
Ω−1+c
.
2
Note that Ω ≥ p1 , then p2 ≥ c − 1 and so 0 ≤ c − p2 ≤
1 by (OA.20).
Since p2 < pb , in period 3 type-S farmers accept the contract, thus
p3 = Ω − 1 + c − αpb − (1 − α)p2 = α(Ω − pb + c − 1) + (1 − α)p0 ≥ c − 1
where the last inequality holds because Ω ≥ p1 ≥ pb and p0 ≥ c − 1. Therefore, we have 0 ≤ c − p3 ≤ 1
by (OA.20).
If p3 < pb , then we are done. Suppose p3 ≥ pb . Then
p4 = Ω − 1 + F (c − p3 ) = Ω − 1 + c − p3 = (1 − α)(Ω − 1 + c − p1 ) + αpb < pb
13
where the last inequality holds because Ω − 1 + c − p1 = p2 < pb . And
p4 = (1 − α)(Ω − p1 ) + (1 − α)(c − 1) + αpb ≥ c − 1
because Ω − p1 ≥ 0 and pb ≥ c − 1.
Then,
p5 = Ω − 1 + c − αpb − (1 − α)p4 = α(Ω − 1 + c − pb ) + (1 − α)p3
= (2α − α2 )(Ω − 1 + c − pb ) + (1 − α)2 p0 ≥ c − 1
because Ω ≥ p1 ≥ pb and p0 ≥ c − 1.
Continuing in this fashion, we obtain that if p2i−1 ≥ pb and p2i−1 > p̄,
p2i = Ω − 1 + c − p2i−1 < pb ,
p2i+1 = [1 − (1 − α)i ](Ω − 1 + c − pb ) + (1 − α)i p0 .
If pb > p̄, then Ω − 1 + c − pb < pb , and we see that there must exist a i2 such that p2i2 < pb and
p2i2 +1 < pb . Then the analysis follows the Subcase 1 above. If pb = p̄, then Ω − 1 + c − pb = p̄. We
could see that limi→∞ p2i = limi→∞ p2i+1 = Ω − 1 + c − p̄ = p̄ =
Ω−1+c−αpb
.
2−α
If p2i−1 = pb = p̄, then Ω − 1 + c − pb = p̄. Thus pt = Ω − 1 + c − pt−1 = p̄ =
Ω−1+c−αpb
2−α
for any
t ≥ 2i − 1.
Case 2: Suppose p0 ≥ pb .
If p0 = p̄, then it must be that p0 = pb and pb = p̄. In this case, we have pt = p̄ for any t ≥ 1. That
is, the price converges to
Ω−1+c
2
=
Ω−1+c−αpb
.
2−α
Hence we just need to study if p0 > p̄.
Since p0 ≥ pb , no farmers accept the contract. Thus
p1 = Ω − 1 + c − p0 < p̄ ≤ pb .
If p2 ≥ pb , then the analysis follows the Subcase 2 of Case 1. If p2 < pb , then the analysis follows
the Subcase 1 of Case 1.
This completes the proof. Proof of Proposition 7.
We prove Proposition 7 for an arbitrary value of p0 , no matter whether
p0 > p̄ or p0 < p̄.
Part 1: We will show the price fluctuates without convergence if pb < p̄.
Without loss of generality, we assume p0 ≤ p̄. (If p0 > p̄, it follows by pb < p̄ that p0 > pb , and so
no farmers accept the contract in period 1. As in the base model, p1 = Ω − 1 + c − p0 < p̄, and the
analysis follows the same process.)
14
If pb ≤ p0 , by farmers’ incentive comparable constraint, no farmers will accept the contract in
period 1. Then p1 = Ω − 1 + c − p0 > pb . Hence, no farmers will accept the contract in period 2.
Continuing in this fashion for t ≥ 2, we find that as in the base model, the price does not converge
and it alternates between two prices.
Next we study the case if pb > p0 . To see whether pt converges, we will first show p2i = pb − (1 −
α)i (pb − p0 ) by induction.
First, we prepare some results that will be used later. Observe that
p0 < pb < p̄ < c.
(OA.22)
By (17) and Assumption 2(a), we have
Ω − 1 ≤ pt ≤ Ω
for any t ≥ 0.
(OA.23)
Since p0 ≥ Ω − 1 and p̄ < Ω, it follows by (OA.22) that
Ω − 1 ≤ pb ≤ Ω.
(OA.24)
Now we derive p2i when i = 1. By Assumptions 4 and 2(b), we have 0 ≤ c − pb ≤ 1 and 0 ≤ c − p0 ≤
1. Thus,
p1 = Ω − 1 + c − αpb − (1 − α)p0
(Ω − 1 − αpb − (1 − α)p0 ) + c ≤ c
by pb ≥ Ω − 1 and pt ≥ Ω − 1 for any t
=
(Ω − αpb − (1 − α)p0 ) + c − 1 ≥ c − 1 by pb ≤ Ω and pt ≤ Ω for any t
Moreover,
p1 − pb = Ω − 1 + c − αpb − (1 − α)p0 − pb = 2p̄ − (1 + α)pb − (1 − α)p0 > 0
where the inequality holds because of (OA.22). Hence, in period 2 no farmers accept the contract
and thus by (17)
p2 = Ω − 1 + F (c − p1 ) = Ω − 1 + c − p1
b
αp + (1 − α)p0 < pb < c by (OA.22)
=
αpb + (1 − α)p0 ≥ c − 1 by Assumptions 2(b) and 4
This established that p2 = pb − (1 − α)(pb − p0 ) and c − 1 ≤ p2 < pb < c.
Suppose p2j = pb − (1 − α)j (pb − p0 ) and c − 1 ≤ p2j < pb ≤ c. Since p2j < pb , type-S farmers accept
the contract in period 2i + 1. Thus
p2j+1 = Ω − 1 + c − αpb − (1 − α)p2j = 2p̄ − pb + (1 − α)j+1 (pb − p0 ) > p̄ > pb
15
where the last two inequalities hold by p0 < pb < p̄. Moreover, it can be checked that c − 1 ≤ p2j+1 ≤ c
because of (OA.23) and (OA.24).
Since p2j+1 > pb , no farmers accept the contract. Thus
p2(j+1) = Ω − 1 + F (c − p2j+1 ) = Ω − 1 + c − p2j+1
(OA.25)
= pb − (1 − α)j+1 (pb − p0 ) < pb
where the last inequality holds because pb > p0 . Moreover, c − 1 ≤ p2(j+1) ≤ c follows because of
(OA.23) and (OA.25).
Hence, by induction we obtain for any i ≥ 1,
p2i = αpb + (1 − α)p2(i−1) = pb − (1 − α)i (pb − p0 ) → pb
as i → ∞.
Since p2i < pb , we have
p2i+1 = Ω − 1 + α(c − pb ) + (1 − α)(c − p2i ) → Ω − 1 + c − pb
as i → ∞.
Consequently, the price does not converge.
Part 2: We show if pb > p̊ and Ω ≥ c or p̄ ≤ pb ≤ p̊, then the price converges to
Ω−1+c−αpb
.
2−α
Combining Lemmas OA.5 and OA.6, we could see the convergence.
Part 3: We show if pb > p̊ and Ω < c, then pt = Ω + α(c − 1 − pb ) for any t ≥ 2.
We first show c − p1 > 1 which will be used to derive p2 . Since pb > p̊, we have
Ω − αpb < (1 − α)(c − 1).
(OA.26)
Observe that
Ω − p̊ = Ω −
Ω − (1 − α)(c − 1) 1
= (1 − α)(c − 1 − Ω) < 0.
α
α
Note also that p0 ≤ Ω and p̊ < pb , so p0 < pb . Thus, type-S farmers accept the contract in period 1.
We have
p1 = Ω − 1 + c − αpb − (1 − α)p0 < c − 1 + (1 − α)(c − 1 − p0 ) ≤ c − 1,
where the first inequality holds because of (OA.26) and the second inequality holds because c − 1 ≤
p0 . Therefore, c − p1 > 1. Moreover, p1 < pb because c − 1 ≤ pb .
Now we are ready to show by induction that pt = Ω − αpb + α(c − 1) for any t ≥ 2.
When t = 2, since p1 < pb , type-S farmers accept the contract in period 2. Thus
p2 = Ω − 1 + α(c − pb ) + (1 − α)F (c − p1 ) = Ω − 1 + α(c − pb ) + 1 − α
16
= Ω − αpb + α(c − 1) < c − 1.
The second equality above holds because c − p1 > 1 and the last inequality above holds because of
(OA.26). Again, p2 < pb because c − 1 ≤ pb .
For any m ≥ 3, suppose pm−1 = Ω − αpb + α(c − 1), pm−1 < c − 1, and pm−1 < pb . To complete the
proof, it suffices to show they all hold for m. In period m, type-S farmers accept the contract and
pm = Ω − 1 + α(c − pb ) + (1 − α)F (c − pm−1 ) = Ω − 1 + α(c − pb ) + 1 − α = Ω − αpb + α(c − 1).
And pm < (1 − α)(c − 1) + α(c − 1) = c − 1 where the inequality holds because of (OA.26). Moreover,
pm < pb because c − 1 ≤ pb .
This completes the proof of Part 3.
Proof of Proposition 9.
By Proposition 8, we know limt→∞ pt = p̄. Hence, the proof follows the
same as that of Lemma 1.
Proof of Lemma 4.
In each cycle i ≥ 1, the prices are as follows,
Ω−1+c
+ (1 − α)i p0 ,
2
p2i+1 = Ω − 1 + c − p2i .
p2i = [1 − (1 − α)i ]
Observe that for each i ≥ 1,
p0 > p2i >
Ω−1+c
= p̄ = pb > p2i+1 > Ω − 1 + c − p0 .
2
(OA.27)
(a) If c − δ ≤ Ω − 1 + c − p0 , then
ub2i = pb − (c − δ) > Ω − 1 + c − p0 − (c − δ) ≥ 0
un2i = p2i−1 − (c − δ) > Ω − 1 + c − p0 − (c − δ) ≥ 0
ub2i+1 = un2i+1 = p2i − (c − δ) > Ω − 1 + c − p0 − (c − δ) ≥ 0.
Thus,
1 s
s
w̄is (δ) = (w2i
+ w2i+1
)=
2
1 n
n
w̄in (δ) = (w2i
+ w2i+1
)=
2
1 b
pb + p2i+1
[p − (c − δ) + p2i+1 − (c − δ)] =
− (c − δ)
2
2
1
p2i + p2i+1
[p2i − (c − δ) + p2i+1 − (c − δ)] =
− (c − δ) = p̄ − (c − δ) > w̄is (δ).
2
2
Proposition 2 gives w(δ) = p̄ − (c − δ) in this case. Clearly,
w̄in (δ) = w(δ) > w̄is (δ).
17
(b) If 2p̄ − p0 < c − δ ≤ p̄, then
ub2i = pb − (c − δ) = p̄ − (c − δ) ≥ 0,
≥ 0 for i ≥ i(δ)
un2i = p2i−1 − (c − δ)
< 0 otherwise
ub2i+1 = un2i+1 = p2i − (c − δ) > p̄ − (c − δ) ≥ 0.
Thus,
1 s
pb + p2i+1
s
w̄is (δ) = (w2i
+ w2i+1
)=
− (c − δ)
2
2
1 n
p̄ − (c − δ)
n
w̄in (δ) = (w2i
+ w2i+1
)= 1
[p
− (c − δ)]
2
2 2i+1
for i ≥ i(δ)
otherwise
Proposition 2 gives w(δ) = 21 [Ω − 1 + c − p0 − (c − δ)] in this case. By (OA.27), we see
min{w̄is (δ), w̄in (δ)} > w(δ). Moreover, it can be checked that w̄in (δ) < w̄is (δ) for i < i(δ) and vice
versa.
(c) If p̄ < c − δ ≤ p0 , then
ub2i = pb − (c − δ) = p̄ − (c − δ) < 0
un2i = p2i−1 − (c − δ) < p̄ − (c − δ) < 0
≥ 0 for c − δ ≤ p2i
b
n
u2i+1 = u2i+1 = p2i − (c − δ)
< 0 otherwise
Thus,
w̄is (δ)
=
w̄in (δ)
1
=
2
[p2i+1 − (c − δ)]
0
for c − δ ≤ p2i
otherwise
Proposition 2 gives w(δ) = 12 [Ω − 1 + c − p0 − (c − δ)] < 0 in this case. Since p2i+1 > Ω − 1 + c − p0 ,
we obtain
w̄is (δ) = w̄in (δ) > w(δ).
(d) If c − δ > p0 , then
ub2i = pb − (c − δ) < p0 − (c − δ) < 0
un2i = p2i−1 − (c − δ) < p0 − (c − δ) < 0
ub2i+1 = un2i+1 = p2i − (c − δ) < p0 − (c − δ) < 0.
Thus, w̄is (δ) = w̄in (δ) = 0. Proposition 2 gives w(δ) = 0 in this case. Therefore,
w̄is (δ) = w̄in (δ) = w(δ).
18
Proof of Proposition 13.
Since pb = p̄, the market dynamics is the same as that in Proposition 8.
Part 1: We will prove f i < f n if i > i0 .
If i > i0 , then
1
b
b
f i = {pb q2i
+ p2i (d − q2i
) + p2i+1 d}
2
1
= {pb α(1 − c + pb ) + p2i [d − α(1 − c + pb )] + (Ω − 1 + c − p2i )d}
2
1
= {pb α(1 − c + pb ) − p2i α(1 − c + pb ) + d(Ω − 1 + c)}.
2
Taking derivative of f i with respect to p2i yields
1
1 b
df i
= − α(1 − c + pb ) = − q2i
< 0.
dp2i
2
2
Note that p2i is decreasing in i and converges to
Ω−1+c
,
2
so f i is increasing in i. Thus
1
d
f i < lim f i = {p̄α(1 − c + p̄) − p̄α(1 − c + p̄) + d(Ω − 1 + c)} = (Ω − 1 + c).
i→∞
2
2
Therefore,
d
1
f i − f n < (Ω − 1 + c) − {p0 (Ω − p0 ) + pm [d − (Ω − p0 )] + (Ω − 1 + c − p0 )d}
2
2
1
m
= − (p0 − p )(Ω − p0 − d) < 0,
2
where the last inequality holds because p0 < pm and Ω − p0 < d.
Part 2: We will prove f i < f n if i < i0 and Ω − 2p0 + pm − d > 0.
If i < i0 , then
1
b
n
b
n
+ p2i q2i
+ pm (d − q2i
− q2i
) + p2i+1 d}
f i = {pb q2i
2
1
b
n
b
n
< {p2i q2i
+ p2i q2i
+ pm (d − q2i
− q2i
) + p2i+1 d}
2
1
b
n
b
n
= {p2i (q2i
+ q2i
) + pm (d − q2i
− q2i
) + (Ω − 1 + c − p2i )d}
2
1
= {p2i (Ω − p2i ) + pm [d − (Ω − p2i )] + (Ω − 1 + c − p2i )d}.
2
[by pb = p̄ < p2i ]
Comparing f i with f n ,
1
f i − f n < {p2i (Ω − p2i ) + pm [d − (Ω − p2i )] + (Ω − 1 + c − p2i )d}
2
1
− {p0 (Ω − p0 ) + pm [d − (Ω − p0 )] + (Ω − 1 + c − p0 )d}
2
1
= (p2i − p0 )(Ω − p2i − p0 + pm − d) < 0,
2
19
where the last inequality holds because p2i < p0 and Ω − p2i − p0 + pm − d > Ω − 2p0 + pm − d > 0.
We can now combine Parts 1 and 2 to see that Part (a) of this Proposition holds.
Part 3: We will show limi→∞ f i is the lower bound.
Recall Part 1 shows limi→∞ f i = d2 (Ω − 1 + c).
By Proposition 7, with contract price pb ≥ p̄, the market price converges to
Ω−1+c−αp
2−α
b
≤ p̄ by pb ≥ p̄, we have d < Ω − p̄ ≤ Ω −
Ω−1+c−αp
2−α
b
Ω−1+c−αpb
.
2−α
Since
where the first inequality holds
because of Assumption 5(a). That is, the market quantity can satisfy the firm’s demand. Thus, the
procurement cost in stabilization, denoted by f b , is
f b = α(1 − c + pb )pb + [d − α(1 − c + pb )]
Ω − 1 + c − αpb
.
2−α
Note that we want to show f b ≥ limi→∞ f i . Observe that f|pb b =p̄ = d2 (Ω − 1 + c) = limi→∞ f i . To
complete the proof, it suffices to show f b is increasing in pb . To this end, taking derivative yields
df b
αd
α
= α(1 − c + 2pb ) −
−
[Ω − 2αpb − (1 + α)(1 − c)]
dpb
2−α 2−α
α
=−
[Ω + d + 3(c − 1) − 4pb ]
2−α
α
>−
[Ω + Ω − p̄ + 3(c − 1) − 4p̄]
2−α
α
Ω+1−c
=−
(−
) > 0.
2−α
2
[by d < Ω − p̄ and pb ≥ p̄]
Proof of Proposition 14.
The market clearing prices are
A
pB
pB − pA
t−1 − pt−1 + 1
t +1
)] − α[1 − F ( t
)]
2
2
A
A
pB
pB
t−1 − pt−1 + 1
t − pt + 1
pB
=
Ω
−
(1
−
α)F
(
)
−
αF
(
).
t
2
2
pA
t = Ω − (1 − α)[1 − F (
(OA.28)
(OA.29)
Thus
A
pB
t − pt = 1 − 2(1 − α)F (
A
pB
pB − pA
t−1 − pt−1 + 1
t +1
) − 2αF ( t
).
2
2
(OA.30)
A
B
A
Since 0 ≤ F (·) ≤ 1, we have |pB
i − pi | ≤ 1 for any i ≥ 1. Observe also that |p0 − p0 | ≤ 1, so (OA.30)
reduces to
A
B
A
B
A
pB
t − pt = 1 − (1 − α)(pt−1 − pt−1 + 1) − α(pt − pt + 1).
Therefore,
A
pB
t − pt = −
1−α t B
1−α B
(pt−1 − pA
) (p0 − pA
t−1 ) = (−
0 ).
1+α
1+α
(OA.31)
20
Plugging (OA.31) back into (OA.28) and (OA.29) yields
A
A
1 + (− 1−α
)t−1 (pB
1 + (− 1−α
)t (pB
0 − p0 )
0 − p0 )
1+α
1+α
+α
2
2
1 1 1−α t B
1
= Ω − − (−
) (p0 − pA
− ∆t (α)
0 )=Ω−
2 2 1+α
2
1
pB
+ ∆t (α).
t =Ω−
2
pA
t = Ω − 1 + (1 − α)
(OA.32)
(OA.33)
(a) Putting α = 0 into (OA.32) and (OA.33), we obtain
1 pB
− pA
0
− 0
,
2
2
− pA
1 pB
0
B
+ 0
.
pA
2n+1 = p2n = Ω −
2
2
B
pA
2n = p2n+1 = Ω −
(b) If α > 0, it is easy to check limt→∞ ∆t (α) = 0. Consequently,
1
B
lim pA
t = lim pt = Ω − .
t→∞
t→∞
2
Proof of Lemma 5.
To complete the proof, it suffices to derive the expression of E[γt PtB ] −
E[γt PtA ]. For the ease of exposition, let Y = E[γt PtB ] − E[γt PtA ]. We have three cases.
B
A
Case 1: Suppose |Pt−1
− Pt−1
| ≤ 1.
By this supposition, (27) reduces to
Y +1 B
A
Y = − (1 − α)(Pt−1
− Pt−1
) + α[1 − 2F (
)] E[γt2 ].
2
(OA.34)
If |Y | ≤ 1, then by (OA.34),
B
A
Y = − (1 − α)(Pt−1
− Pt−1
) − αY E[γt2 ]
(1 − α)E[γt2 ] B
A
Y =−
(Pt−1 − Pt−1
).
1 + αE[γt2 ]
Note that
(1−α)E[γt2 ]
1+αE[γt2 ]
≤ 1 because (1 − 2α)E[γt2 ] ≤ 1, so |Y | ≤ 1.
If Y > 1, then by (OA.34),
B
A
Y = [−(1 − α)(Pt−1
− Pt−1
) − α]E[γt2 ] ≤ (1 − α − α)E[γt2 ] = (1 − 2α)E[γt2 ] ≤ 1,
B
A
contradicting the assumption that Y > 1. The first inequality above holds because Pt−1
− Pt−1
≥ −1.
If Y < −1, then by (OA.34),
B
A
Y = [−(1 − α)(Pt−1
− Pt−1
) + α]E[γt2 ] ≥ [−(1 − α) + α]E[γt2 ] = −(1 − 2α)E[γt2 ] ≥ −1,
21
B
A
contradicting the assumption that Y < −1. The first inequality above holds because Pt−1
− Pt−1
≤ 1.
B
A
Combining the above three scenarios, we see that if |Pt−1
− Pt−1
| ≤ 1, then it must be that
Y =−
(1 − α)E[γt2 ] B
A
(Pt−1 − Pt−1
).
1 + αE[γt2 ]
Plugging back into (26) yields
PtB − PtA = −
(1 − α)γt
A
(P B − Pt−1
).
1 + αE[γt2 ] t−1
B
A
− Pt−1
> 1.
Case 2: Suppose Pt−1
By the supposition, (27) reduces to
Y +1 Y = − (1 − α) + α[1 − 2F (
)] E[γt2 ].
2
(OA.35)
If |Y | ≤ 1, then by (OA.35),
Y = − (1 − α) − αY E[γt2 ]
(1 − α)E[γt2 ]
Y =−
.
1 + αE[γt2 ]
Since (1 − 2α)E[γt2 ] ≤ 1, |Y | ≤ 1.
If Y > 1, then Y = [−(1 − α) − α]E[γt2 ] = −E[γt2 ] < 1, contradicting the assumption that Y > 1.
If Y < −1, then Y = [−(1 − α) + α]E[γt2 ] = −(1 − 2α)E[γt2 ] ≥ −1, contradicting the assumption
that Y < −1.
B
A
Taking the above three scenarios into consideration, we see that if Pt−1
− Pt−1
> 1, then it must
be
Y =−
(1 − α)E[γt2 ]
.
1 + αE[γt2 ]
Plugging back into (26) yields
PtB − PtA = −
(1 − α)γt
.
1 + αE[γt2 ]
B
A
Case 3: Suppose Pt−1
− Pt−1
< −1. Symmetric to Case 2.
Part of Proof of Proposition 15.
Observing (24) and (25), in order to see whether PtA and PtB
converge, it is essential to study whether PtB − PtA converges.
Part 1: We will prove the convergence of PtB − PtA if I(α, γ) = 0 and non-convergence if I(α, γ) > 0.
Define the following iteration:
(1 − α)γt eB
A
PetB − PetA = −
(P − Pet−1
),
1 + αE[γt2 ] t−1
22
|Pe0B − Pe0A | ≤ 1.
In preparation of analyzing PtB − PtA , we derive some useful results about PetB − PetA . We have
(1 − α)γt−1
(1 − α)γ1 eB eA
(1 − α)γt
·
...
|P − P0 |.
|PetB − PetA | =
2
2
1 + αE[γt ] 1 + αE[γt−1 ]
1 + αE[γ12 ] 0
Thus,
ln |PetB − PetA | =
t
X
ln
i=1
(1 − α)γi
+ ln |Pe0B − Pe0A |.
1 + αE[γi2 ]
Through some algebra, we get
Pt
(1−α)γi
(1−α)γi
ln |PetB − PetA | − ln |Pe0B − Pe0A | − tE[ln 1+αE[γ
2] ]
i=1 [ln γi − E[ln 1+αE[γ 2 ] ]]
i
i
q
q
=
−→ N (0, 1).
(1−α)γi
(1−α)γi
t · var[ln 1+αE[γ 2 ] ]
t · var[ln 1+αE[γ 2 ] ]
i
i
Then we have
lim P r(|PetB − PetA | < ) = lim P r(ln |PetB − PetA | < ln )
t→∞
t→∞
(1−α)γi
(1−α)γi
ln |PetB − PetA | − ln |Pe0B − Pe0A | − tE[ln 1+αE[γ
ln − ln |Pe0B − Pe0A | − tE[ln 1+αE[γ
2 ]
2 ]
i]
i]
q
q
= lim P r(
<
)
(1−α)γi
(1−α)γi
t→∞
t · var[ln 1+αE[γ
]
t
·
var[ln
]
2
1+αE[γi2 ]
i]
Pt
(1−α)γi
(1−α)γ
(1−α)γi
ln − ln |Pe0B − Pe0A | − tE[ln 1+αE[γ 2i ] ]
i=1 [ln 1+αE[γ 2 ] − E[ln 1+αE[γ 2 ] ]]
i
i
i
q
q
= lim P r(
<
)
(1−α)γi
(1−α)γi
t→∞
t · var[ln 1+αE[γ 2 ] ]
t · var[ln 1+αE[γ 2 ] ]
i
i
Pt
(1−α)γi
(1−α)γi
s
i=1 [ln 1+αE[γ 2 ] − E[ln 1+αE[γ 2 ] ]]
ln − ln |Pe0B − Pe0A |
(1 − α)γi
t
i
i
q
= lim P r(
<q
−
E[ln
])
(1−α)γi
(1−α)γi
(1−α)γi
t→∞
1
+ αE[γi2 ]
var[ln
2 ]
]
]
t · var[ln
t · var[ln
1+αE[γi2 ]
1+αE[γi ]
1+αE[γi2 ]
Hence,
(
lim
t→∞
P r(|PetB
− PetA |
< ) =
(1−α)γi
P r(N (0, 1) < −∞) = 0 if E[ln 1+αE[γ
2 ] ] = I(α, γ) > 0
i
(1−α)γi
P r(N (0, 1) < 0) = 1/2 if E[ln 1+αE[γ
2 ] ] = I(α, γ) = 0.
i
By the same process, we could show
(
lim P (|PetB − PetA | > 1) =
t→∞
1
(1−α)γi
if E[ln 1+αE[γ
2 ] ] = I(α, γ) > 0
i
(1−α)γi
1/2 if E[ln 1+αE[γ
2 ] ] = I(α, γ) = 0.
i
Observing that PtB − PtA , within the range of [−1, 1], follows a similar process as PetB − PetA . Once
PtB − PtA jumps out of [−1, 1], it nevertheless will be pulled back into [−1, 1].
If I(α, γ) > 0, then limt→∞ P r(|PetB − PetA | < ) = 0 and limt→∞ P (|PetB − PetA | > 1) = 1, implying
that even if PtB − PtA can be pulled back, the process will inevitably keep growing out of the range
of [−1, 1]. Thus, PtB − PtA will not converge.
23
If I(α, γ) = 0, then limt→∞ P r(|PetB − PetA | < ) = 1/2 and limt→∞ P (|PetB − PetA | > 1) = 1/2, implying that the process PtB − PtA within the range of [−1, 1] will end up with two possible results: one
is out of the range of [−1, 1] and the other is 0, each with half probability. Since the process will
be pulled back whenever it gets out of [−1, 1] and there is 1/2 probability that it will converge to 0
whenever it is in [−1, 1], the process will eventually converge to 0. That is, limt→∞ P r(|PtB − PtA | <
) = 1 for any > 0.
Part 2: We analyze whether PtA and PtB converge.
We have if I(α, γ) = 0, then the process PtB − PtA will converge to 0. It now follows by (26) that
limt→∞ P r(|E[PtB γt ] − E[PtA γt ]| < ) = 1. Plugging back into (24) and (25) yields
lim P r(|PtA − P̄ | > ) = lim P r(|PtB − P̄ | > ) = 0.
t→∞
t→∞
If I(α, γ) > 0, then PtB − PtA will not converge. Therefore, by (24) and (25) the prices (PtA , PtB )
will not converge neither.