Cooperation, Cheating, and the Competitive Yardstick

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professor in the Department of Mathematics, at Oregon State University. ... cheating, by members of the now bankrupt Tri Valley Growers, formerly the largest.

Cooperation, Cheating, and the Competitive Yardstick

Robin M. Cross, Steven T. Buccola, and Enrique A. Thomann

Robin M. Cross is a research associate and Steven T. Buccola a professor in the Department of Agricultural and Resource Economics, and Enrique A. Thomann is a professor in the Department of Mathematics, at Oregon State University. Authors are available by e-mail at [email protected] or phone at (541) 737-1397.

An earlier version of this paper was presented at the American Agricultural Economics Association Annual Meeting, Long Beach, California, July 23-26, 2006. Research supported by Rural Business-Cooperatives Service (RBS), USDA, Research Agreement No. RBS-01-09. RBS/USDA is not responsible for any views expressed in this article.

Cooperation, Cheating, and the Competitive Yardstick

Robin M. Cross, Steven T. Buccola, and Enrique A. Thomann

Abstract Farmer-members of agricultural processing cooperatives typically sign marketing contracts not only with their own cooperative but with investor-owned (IOF) processors as well. Cooperative contracts usually are agreements to sell the produce of a given acreage while IOF contracts are agreements to sell a given tonnage, and farmers are widely known to illicitly under- or over-supply the cooperative to take advantage of harvest conditions after the contracts have been signed. We characterize such cheating phenomena as rational for both the farmer and cooperative. In particular, we derive the value of the option to illicitly shift produce and show that, in contrast to Nourse’s competitive yardstick hypothesis, it increases as IOF competition intensifies. A fairly priced shifting option is Pareto improving, increasing grower returns, lowering cooperative per-unit costs, and reducing delivery shortfalls at no additional per-unit cost to the investor-owned firm.

Key words:

agricultural cooperatives, fraud, processing tomatoes, production contracts

Cooperation, Cheating, and the Competitive Yardstick

Under intense competition from investor-owned firms (IOF’s), agricultural cooperatives frequently struggle to control member tendencies to over- or under-deliver contracted produce, that is to cheat. When crop yields are high and exceed the amounts contracted for sale to IOF processors, some members shift excess produce from the IOF to the more lenient cooperative, thereby receiving payment for produce that otherwise would be plowed under. When yields are low, members may shift produce away from the cooperative in order to fulfill their higher paying IOF contracts. An “extra row” of farm produce represents a low percentage of contracted acreage and is difficult to discern amid harvest activity. Enforcing cooperative delivery terms under these circumstances is costly and often conflicts with the cooperative philosophy of providing a “home” for member produce. Recent analytical treatments of fraud activity have focused on detecting fraudulent crop insurance claims (Atwood, Robinson-Cox, and Shaik 2006), designing compliance incentives for conservation programs (Giannakas and Kaplan 2005) and pollution permit markets (Milak 2006), and determining optimal tax enforcement strategies (Peralta et al. 2006). We instead examine the case in which the fraudulent activity is conducted by the firm’s owners. The market we consider is one in which competing investor-owned production contracts are available and an organized commodity exchange for the commodities involved is not available.

Nourse’s (1945) competitive yardstick hypothesis suggests that as competition intensifies, the incentive to cooperate wanes. The possibility of cheating is, at least informally, built into most processing cooperative’s member marketing contracts. That possibility has value which can be formalized and enumerated. We examine the optionto-cheat and show, in contrast to Nourse, that its value strictly increases as competition intensifies. To illustrate this result, we consider an alleged case of harvest-shifting, or cheating, by members of the now bankrupt Tri Valley Growers, formerly the largest agricultural processing cooperative in the United States. Tri Valley competed for raw tomatoes with investor-owned firms, including Hunts, Heinz, Campbell, Ragu, and Del Monte. The IOFs together controlled 56% of the California processing-tomato market. No organized cash exchanges existed in any of the specialty fruit and vegetable markets in which Tri Valley operated, including the processing-tomato market. Because of tomatoes’ high weight-to-value and perishable nature, 99% of California’s processing tomatoes were, according to the California Tomato Growers Association (CTGA), planted under contract in the 1991 – 2000 period. For this reason, and like many processing cooperatives, Tri Valley struggled with member cheating and experimented with a range of enforcement and monitoring policies. We use information from internal cooperative records, as well as county and competitor contract data, to compare BlackScholes-style fair market contract prices with and without cheating. We find the option-to-cheat, when fairly priced, to be a Pareto-improving market mechanism. Like a small personal commodity exchange, the option-to-cheat provides

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growers with a finite crop reserve when yields are low, and a destination for excess production when yields are high. Cheating growers deliver more produce and at higher average prices than do non-cheating growers, so their net farm returns are higher as well. These higher grower returns allow cooperatives to reduce the forward prices they charge for their processed products. IOF processors who purchase raw product from cooperative cheaters experience fewer contract shortfalls, and face lower delivery volatility at no higher per-unit cost, than do those who don’t so purchase. Formally incorporated into the cooperative contract, the cheating option transforms the cooperative from IOF rival to beneficial partner. Economic Model Consider an economy with two processors – a cooperative and an investor-owned firm – each of which purchase raw product from a given set of growers. The IOF processor has monopoloid power in its finished goods markets, while the cooperative delivers to a separate and competitive market. As a raw product buyer capable of scaling its purchase activity to market conditions, the cooperative poses a credible threat to the IOF processor, enabling grower members to extract any IOF monopolistic rent. Assume there is no organized commodity exchange. Growers forward contract all their acreage, half with the cooperative and half with the IOF. For each contracted acre, the cooperative member receives the relatively low cooperative price p1 for each ton y of harvested produce. The grower’s cooperative-contract revenue function r1 then can be written

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r1 =

(1)

p1 y .

In contrast, the IOF grower receives a relatively high IOF per-ton price p2 for the lesser of harvested produce y or delivery threshold y D : (2)

r2

=

p2 min { y ,

yD} .

Figure 1 illustrates the per-acre contract revenues associated with each of the cooperative and IOF contracts over a range of farm per-acre yields y . For illustration, the cooperative in figure 1 pays p1 = $1 per unit harvested. The IOF contract pays p2 = $2 per unit up to a threshold of y D = 5 units, and nothing thereafter. Combined revenue from the grower’s portfolio of two contracts – equivalently from two acres – therefore is the sum of the two payoffs r1 + r2 . Consider now a grower willing and able ex post to shift some proportion δ of the harvest from one contract to the other. Let shift proportion δ be expressed as a percentage of total per-acre agricultural yield y , taking any value in the interval [ −α , α ] where α is a positive constant. This interval may represent the grower’s ethical limit, a formal or tacit allowance from the contracting firm, or the threshold below which detection will not occur. The cheating grower then chooses the proportion of raw product that, if shifted, maximizes total portfolio payoff. Denote the cooperative and IOF revenue functions in these cheating situations as r1 (α ) and r2 (α ) respectively. The upper-most dashed line in figure 2 illustrates combined revenue r1 (α ) + r2 (α ) in the presence of cheating. This function has three kinks: original delivery

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threshold y D , and two more determined by the magnitude of cheating parameter α . When terminal yield exactly matches IOF delivery threshold y = y D , cheating incentives go to zero and cheating revenues become identical to those with no cheating. This is because, since the IOF pays nothing for excess production, a one-unit shift from the cooperative to the IOF brings a $1 loss in return for $0 gain. Alternatively, a one-unit shift from the IOF to the cooperative brings a $2 loss in return for a $1 gain, the cooperative’s low price per unit. Therefore again, no shift takes place. When terminal yield coincides with the lower kink at y = y D /(1 + α ) , the grower shifts enough produce from the cooperative to fulfill the IOF contract, namely the maximum allowable shift δ = - α . This shift reduces cooperative contract revenue and elevates IOF contract revenue, as indicated by the two dashed lines. Finally, the lower-most dashed line representing net cheating revenue z (α ) in figure 2 represents the additional revenue attributable to the option-to-cheat – namely the difference between the portfolio payoff with and without cheating: (3)

z (α ) = r1 (α ) + r2 (α ) − r1

− r2 ,

Note that z (α ) is everywhere non-negative. Faced with a non-negative payoff and strictly positive expected value, the grower “cannot lose” by cheating. This is a key characteristic of a true option. Cheating represents not a single option, but a range of harvest-shift possibilities, each dependent on per-acre yield y and the magnitude α of the shift. See the Appendix for the general specification of z (α ) .

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Both the cooperative and IOF processor require their farmer-suppliers to follow prescribed production practices. Some of these practices, such as varietal choice and planting date, bear costs that are known at the beginning of the growing season. Others, such as irrigation methods and chemical use, depend on conditions that evolve throughout the growing season. The latter affect per-acre yields and input costs jointly. For instance, rising temperatures simultaneously may promote plant growth and require additional irrigation water. The grower’s expected total input cost c is the sum of fixed costs a and per-ton variable costs b :

c = a + b y,

(4)

where y is expected per-acre yield at harvest time. Grower’s net contract payoff from goods shipped to the cooperative is, then, difference r1 − c between cooperative-contract revenue and cost. Stiglitz’ (1974) classic paper on wages and rents employed a similar linear production contract structure to analyze incentives and risk sharing, focusing on a sharecropping model that included variable delivery quantities and fixed input controls. In his analysis of efficiency gains under a Soviet management incentive model, Weitzman (1976) used a linear production contract with budget constraints. Linear contracts are ubiquitous to agriculture, encompassing broilers (Goodhue 2000), hogs (Key and McBride 2003) , tomatoes (Hueth and Ligon 2002), wine grapes (Goodhue, et al. 2003), and agricultural cooperatives (Bogetoft 2005). Other production contract mechanisms relevant to equation (4) have been examined in an agricultural context, such as

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enforcement (Levy and Vukina 2002, Kvaløy 2006) and input control (Goodhue 1999). Hueth and Ligon (2002) discuss quality premia variations in processing tomato contracts. We assume quality provisions are identical across contracts. Fair-Market Cheating Values

Two possible notions of “value” may be applied to the option-to-cheat. For the cooperative firm, the option may lower the required forward price, sometimes called a rebalanced forward contract. For the cheater, the option leads to “extra” revenue, a home for surplus production when agricultural yields are high. We are interested in both such notions, and will explore the former here and the latter in our discussion of the empirical results. To determine whether the cooperative’s required forward price declines when its members are allowed to cheat, we use a Black-Scholes (1973) fair-market pricing model. The Black-Scholes price is “fair” because it provides neither farmer nor processor with an “arbitrage opportunity,” that is, the opportunity simultaneously to buy an asset at one price and sell it at another, obtaining a riskless profit. The Black-Scholes model has been shown consistent with a broad class of Radner-type economic equilibria (Kreps 1981), and is independent of agent utility functional form and processor cost structure. Lacking a liquid cash exchange, the discount rate is recovered directly from the reference IOF contract (Björk 1998, Alaton, et al. 2002). As a way of seeing that the forward price would indeed fall if cheating were permitted, observe that the option’s contribution value or “premium” must be positive ex

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ante because, as illustrated in figure 2, its payoff is positive for the cheater over all possible outcomes. The option’s value can be expressed in terms of expected value (Cox, Ross, and Rubenstein, 1979). From the cheater’s perspective, this value is (5)

E ρ ( z (α ) ) > 0 ,

where E ρ is the conditional expectation operator given discount rate ρ . Denote the fair-market forward price in the presence of cheating as p1∗ (α ) , and in the absence of cheating as p1∗ . By definition, a forward contract includes no initial premium payment. The forward price must therefore be set to a level bringing zero value

ex ante to both cooperative and member. When cheating is present, fair-market cooperative forward price p1∗ (α ) must solve (6)

E ρ ( r1 (α ) − c ) = 0 .

In the absence of cheating, forward price p1∗ must solve (7)

E ρ ( r1 − c ) = 0 .

Cheating lowers the required cooperative forward price if p1∗ (α ) < p1∗ . Because no convenient analytical expression exists for the density of the arithmetic average of a log-normal random variable (Kemna and Vorst 1990), the average term in equation (4), we obtain solution values via standard Monte Carlo simulation (Boyle, Brodie, and Glasserman 1997). Monte Carlo simulation is consistent, asymptotically normal, and accurate to an arbitrary level depending on the number of

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draws (Campbell 1997). We also use simulation for mean deliveries and delivery variance estimates with and without cheating. Yield volatility is a key parameter in the Black-Scholes model. We estimate this statistic via Maximum Likelihood Estimation (MLE). MLE is consistent and asymptotically efficient and tends to be more efficient in small samples than GMM when observations are independently and identically distributed and when the distributional choice is correct. When the distributional choice is incorrect, parameter solutions may be Quasi-Maximum Likelihood estimates (QMLE). QMLE solutions retain consistency under certain regularity conditions but are less efficient than MLE in finite samples (White 1982). Statics

In order to test Nourse’s (1992) competitive yardstick hypothesis, one needs to know how the cheating option value would change as competition intensifies, that is as the IOF’s forward purchase price is driven upward by either a rival cooperative, collective bargaining, or competing IOFs. Cheating exploits price differentials among contracts. Therefore, forces such as IOF competition which increase these differentials enhance cheating’s value. More formally, consider the non-trivial case in which at least some cheating occurs, positive delivery is accepted under the IOF contract, and expected peracre yields are nonzero. Under these conditions, we offer our central claim.

Proposition: The cooperative member’s cheating option gains value as competition intensifies:

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∂ E ρ ( z (α ) )

(8)

∂p2

> 0,

where p2 is the IOF forward price, and α , y D , E ( y ) > 0. Proof follows directly from the nature of this first-order condition of inequality (5) and is provided in the Appendix. Condition values are illustrated in the Results section over a range of cheating and threshold levels. Two other economic forces affect the value of the cheating option: technical innovation and capital market maturation. Technical innovation, such as GMO technologies for developing new plant varieties, improves agricultural productivity and reduces production risk. The first-order condition of (5) with respect to production risk – in our case per-acre yield volatility σ – is given by ∂ E ( z (α ) )

(9)

∂σ

.

A positive value of (9) suggests volatility-reducing innovation diminishes the need for, and hence value of, the cheating option. We find below that such is the case. The cooperative is, because of its ability to retain equity from member patronage, uniquely suited to subscribe new equity in the face of thin capital markets and high interest rates (Cross and Buccola 2004). As capital markets mature, however, interest rates fall and risk premia narrow. It therefore is particularly important to understand how cheating option values change when capital markets develop. If their values decline, that is if the value of cheating wanes, we have

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(10)

∂ E ( z (α ) ) ∂ρ

> 0.

This first-order condition also is illustrated below. Data

An instance of alleged cheating occurred at Tri Valley Growers, formerly the largest cooperative agricultural processor in the United States. We examine the 20-year period 1977 - 1996, before financial difficulties at Tri Valley were made public. The CTGA publishes annual data on IOF forward tomato prices, which averaged $98 per ton and ranged from $64 to $181 during the study period. Tri Valley forward prices during those same years, taken from the company’s audited Additions to Financial Statements, averaged $99 per ton and ranged between $65 and $180. All values are in constant 2005 dollars. Kern County, one of the California counties in which Tri Valley operated, has published yield statistics since 1963. While widely used in applied work (Ker and Coble 2003), county-level yields are known to be insufficiently disaggregated. In particular, they tend to “average out” some of the volatility experienced at the farm-level and thus depress variance estimates. Kern County’s harvested processing tomato acreage was, for example, 4600 acres in 1996 compared with an average Tri Valley member harvest of 300 to 900 acres. Because option values are positively related to variance, our option value estimates will tend to be understated.

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We will employ production costs in the San Joaquin Valley of California, where double-row direct seeding and transplanting are the two predominant planting methods for processing tomatoes. The University of California Cooperative Extension Service (UCCES) estimates that transplanted acreage reached just under 30% of total California processing tomato acreage by 2004 (Hartz and Miyao 1997, Le Strange 2004). A transplanted acreage estimate of 25% will be assumed for 2002. The UCCES reports fixed production costs of $1,368 per acre and variable costs of $14.82 per ton in that year (Valencia, et al. 2002a, 2002b). Except where otherwise stated, we use a cheating magnitude of α = 0.10 for illustrative purposes. Interviews with cooperative managers suggest cooperative-to-IOF or IOF-to-cooperative harvest shifting varied widely by year and member, as influenced by changing market conditions and cooperative enforcement policies. Results

Figure 3 illustrates cooperative forward tomato purchase prices with and without the option to cheat, as specified respectively in equations (6) and (7). IOF forward purchase prices are shown for comparison. Introducing the cheating option reduces the arbitragefree cooperative forward price by an average 6.7% or $5.60 per ton, and ranges from a high of 8.5% in 1990 to a low of 3.8% in 1994. Rather than formalize the member’s option-to-cheat – and commensurately reduce the forward price – Tri Valley attempted to minimize cheating through a number of conventional monitoring and enforcement policies. To the extent that cheating persisted

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in the face of these efforts, the Tri Valley forward contract was mispriced, failing to reflect the additional value which shifting provided to its members and hence paying “too much” for member raw product. Such inflations in raw product purchase prices likely contributed to Tri Valley’ working-capital shortfalls in the late 1990’s, exacerbating its slide into bankruptcy in 2000. Figure 4 illustrates simulated mean deliveries and delivery volatilities in the cheating and no-cheating cases. Interestingly, cheating boosts average raw product delivery volumes to both the cooperative and IOF processor. Deliveries to the IOF rise because there are fewer contract shortfalls. Deliveries to the cooperative rise because growers shift produce from the IOF to the cooperative contract during surplus years. Cheating also transfers delivery volatility from IOF to cooperative processor. A closer look at the two contracts when no cheating is permitted suggests why a formal cheating provision may be Pareto-improving. When no cheating is allowed, the IOF contract’s basic design leads to more stable, targeted deliveries, and at higher per-unit costs (forward prices), than does the cooperative contract. This is consistent, for example, with our experience that the branded processing tomato markets which Hunts, Heinz, Ragu, Campbell’s, and other IOF’s serve are comparatively high-valued and stable, while the commodity-style markets which Tri Valley served (for example, its large government service contracts) were comparatively low-cost and volatile. Cheating strengthens these relative differences, reducing any IOF raw-product shortfalls and providing a home for the cooperative member’s produce. Reduced delivery volatility and shortfall frequency add, of course, no additional per-unit cost to the IOF. The cooperative’s per-unit costs

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may actually decrease as the cooperative reduces its forward prices to capture member gains from the cheating option. Competition’s impact on the value of cheating, expressed by first-order condition (8), is everywhere positive. As competition between the cooperative and IOF intensifies, the expected value of the cheating option increases, boosting expected member revenue. Figure 5 illustrates the magnitude of competition’s impact on member revenue over a range of delivery thresholds y D and cheating levels α . The graph suggests that competition’s impact is magnified as delivery thresholds and cheating levels rise. This member-benefiting result raises the question why an IOF processor would not itself enter the same low-value, commodity-style output market that cooperatives pursue, internalizing gains from the cheating mechanism and eliminating the cooperative competitor. One factor may be that returns on equity in commodity-style markets are unattractive to IOFs. During the 1977 – 1996 study period, publicly-traded investorowned firms achieved significantly higher and more stable net returns than Tri Valley Growers did (Cross and Buccola 2004). Indeed, Tri Valley tried at one point to enter the high-value market, investing heavily in an unsuccessful branded product line (Hariyoga 2004). To the extent such crossing of product-type lines is unattractive, the cheating mechanism remains a cooperative-provided service. Our last graph, Figure 6, shows that maturing capital markets depress the value of cheating. At all harvest volatility levels, marginal cheating value declines as interest rates ρ fall. The impact on these values, however, of volatility-reducing technological innovation is interest-rate-dependent. At high interest rates, yield stabilization enhances

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the gains from cheating; at low interest rates, it reduces the gains. This is reasonable because the firm requires higher returns to compensate for higher interest rates or higher risk-premia.

Conclusion

We have examined the widespread phenomenon under which a farmer-member of an agricultural processing cooperative surreptitiously shifts raw product deliveries between the cooperatives’ acreage-based delivery contract to an investor-owned firm’s tonnagebased contract, a practice encouraged by the cooperative philosophy of providing a home for member produce. We find that “legalizing” such a practice – in particular, providing a fairly priced option to shift harvest from one contract to another – can be Paretoimproving, transforming the cooperative from IOF-adversary to beneficial partner. Such an option increases the grower’s net returns, lowers the cooperative’s unit costs, and enables the investor-owned firm to avoid production shortfalls at no additional unit cost. In other words, fairly priced cheating options are win-win. Because equity returns typically are lower in member-owned than in investor-owed firms, traditional and newgeneration cooperatives are equally suited to provide these options. And, as rising competition boosts the purchase prices which IOF processors offer their farmer-suppliers, a farmer’s options-to-cheat, and the liquidity it provides, become ever more valuable. Cheating options therefore are likely to gain greater attention as economic globalization intensifies.

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References

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Cross, R., and Buccola, S. "Adapting Cooperative Structure to the New Global Environment." American Journal of Agricultural Economics 86, no. 5(2004): 1254-1261. Giannakas, K., and J. D. Kaplan. "Policy Design and Conservation Compliance on Highly Erodible Lands." Land Economics 81, no. 1(2005): 20-33. Goodhue, R. E. "Broiler Production Contracts as a Multi-Agent Problem: Common Risk, Incentives and Heterogeneity." American Journal of Agricultural Economics 82, no. August(2000): 606-622`. Goodhue, R. E. "Input Control in Agricultural Production Contracts." American Journal of Agricultural Economics 81, no. August(1999): 616-620. Goodhue, R. E., Heien, D. M., Lee, H., and Sumner, D.A. "Contracts and Quality in the California Winegrape Industry." Review of Industrial Organization 23(2003): 267282. Hariyoga, H. "An Economic Analysis of Factors Affecting the Failure of an Agricultural Marketing Cooperative: The Bankruptcy of Tri Valley Growers." Ph.D., University of California Davis, 2004. Hartz, T. K., and Miyao, G. . "Processing Tomato Production in California." University of California, Division of Agriculture and Natural Resources, Publication 7228, 1997. Hueth, B., and Ligon, E. "Estimation of an Efficient Tomato Contract." European Review of Agricultural Economics 29, no. 2(2002): 237-253.

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Appendix General expression for the option-to-cheat [Equation (3)] The cheating option payoff z (α ) in equation (3) is a function of two especially simple contracts, each with a single forward price and payoff threshold and identical input control terms. More complex linear contracts are common, for example the piecewise linear Soviet incentive payment system considered by Weitzman (1976) or the U.S. personal and corporate income tax models, with multiple-brackets (threshold) and corresponding rates (prices). A more general expression for z (α ) , one that accommodates multiple delivery thresholds and prices, requires some additional notation. First, the cooperative forward price can be divided into two prices p11 and p12 , for deliveries below and above the threshold y , respectively. In our cooperative’s case, the two prices are identical p11 = p12 . Similarly, let the IOF pay p12 for deliveries below threshold y and p22 for deliveries above it. Because the IOF in our study pays nothing for surplus production, the second price equal to zero p22 = 0 .

To accommodate multiple delivery thresholds, such as those specified by the U.S. income tax system, consider a contract with J thresholds, and replace the y notation with thresholds k j ∈ ℜ + , j = 1,… , J , to avoid confusion with the agricultural yield variable. Because harvest time is the terminal period T, denote the per-acre yield level at harvest

time yT ∈ℜ+ .

Payoff z (α ) , from shifting deliveries between two forward contracts, is the sum of three functions, z (α ) =

(A.1)

f (α ) +

g (α ) + h (α ) ,

where f (α ) is the sum J +1

∑α

(A.2)

j =1

s1j − s2j yT 1⎡ y ⎣

j⎤ T ∈D1 ⎦

.

Function g (α ) is a sum of maximization problems

J +1

(A.3) ∑ j =1

+ ⎧ j ⎛ 1 ⎞ j⎞ ⎫ j j j +1 j +⎛ ⎪( s1 − s2 )( k − yT ) + (1 + α ) ( s1 − s2 ) ⎜ yT − ⎜ ⎟ k ⎟ ,⎪ ⎝ 1+ α ⎠ ⎠ ⎪ ⎪ ⎝ max ⎨ ⎬ 1⎡ yT ∈D2j ⎤ . ⎦ + +⎛ 1 ⎞ ⎛ ⎞ + j j j j j j 1 ⎪ s − s k − y + (1 + α ) s − s ⎪ ⎣ y k − ( )( ) ( ) T 2 1 ⎜ T ⎜ 1+ α ⎟ ⎟ ⎪ ⎪ 2 1 ⎝ ⎠ ⎠⎭ ⎝ ⎩

Here, the maximal element is determined by the magnitude of the price differentials between contracts. Function h (α ) is given by

J +1

(A.4)

∑ j =1

⎧ j +1 1 ⎞ j ⎞⎫ j +1 j + j +1 j + ⎛⎛ ⎪( s1 − s2 )( yT − k ) + (1 − α ) ( s1 − s2 ) ⎜ ⎜ ⎟ k − yT ⎟ , ⎪ ⎪ ⎝ ⎝ 1−α ⎠ ⎠⎪ max ⎨ ⎬ 1⎡ yT ∈D3j ⎤ . ⎦ ⎪ s j +1 − s j +1 y − k j + + (1 − α ) s j +1 − s j + ⎛ ⎛ 1 ⎞ k j − y ⎞ ⎪ ⎣ ( )( ) ( ) 2 1 2 1 T T ⎜ ⎟ ⎜ ⎟ ⎪ ⎝ ⎝ 1−α ⎠ ⎠ ⎪⎭ ⎩

Here, ( yT − k j ) = max ( yT − k j , 0 ) , and 1⎡ y ∈D j ⎤ is the indicator function, taking the +



T

1



value 1, when per-acre yield is within an interval D1j and zero otherwise. The yield variable takes on values in one of J x 3 intervals, D1j , D2j , and D3j , j = 1,…,J, corresponding to regions of the yield domain:

A.2

(A.5)

D1j

⎡⎛ 1 ⎞ j −1 ⎛ 1 ⎞ j ⎞ = ⎢⎜ ⎟k , ⎜ ⎟k ⎟ ⎝ 1+ α ⎠ ⎠ ⎣⎝ 1 − α ⎠

D2j

⎡⎛ 1 ⎞ j j ⎞ = ⎢⎜ ⎟k , k ⎟ ⎣⎝ 1 + α ⎠ ⎠

D3j

⎡ ⎛ 1 ⎞ j⎞ = ⎢k j , ⎜ ⎟k ⎟ ⎝ 1−α ⎠ ⎠ ⎣

Figure 3 illustrates each of the three regions for the J = 1 case. Regions D1j , j = 1,…,J, correspond to yield values lying between contract thresholds such that no threshold k j lies within interval (1 + δ ) yT for any possible value of δ ∈ [ −α , α ] . Regions D2j correspond to yield values just below thresholds k j such that (1 + α ) yT reaches or exceeds the threshold. Finally, regions D3j correspond to terminal yield values just above thresholds k j such that (1 − α ) yT reaches or exceeds the threshold.

General expression for input cost control [Equation (4)] The simplified cost function in equation (4) is derived from a more general const control expression. Expanding our cost notation somewhat, let the cooperative member agree to follow a set of standard input practices C[0, T ] over the entire production period, from field preparation at time 0 to harvest at time T. We express the member’s costs in terms of rates of investment per expected harvestable ton yt ∈ ℜ+ , over the growing period t ∈ [ 0, T ] . Denote the member’s fixed costs investment rate

a ∈ ℜ , and define variable T

costs in terms of two rates of investment, both dependent on the current growing conditions. When growing conditions are “poor,” that is below some critical threshold

A.3

h ∈ ℜ+ , let members invest at a rate

b ∈ ℜ . When growing conditions are favorable T

yt ≥ h , members may invest at a rate

c ∈ ℜ . For example, members may apply pest T

control inputs at a higher rate when growing conditions are especially warm or wet. As defined, the input cost function is given by T

C[0,T ] = ∫

(A.6)

0

a T

+

b ( yu − h )1[ yu >h] T

+

c ( yu − h )1[ yu ≤h] T

du ,

where 1[ yu ≤ h] is the indicator function, taking the value 1, when per-acre yield is below threshold h and zero otherwise. In the case of a single variable cost rate b = c, the integral passes through the expression, resulting in C[0,T ] = a + b ( y − h ) ,

(A.7)

where y is the arithmetic average T

1 y = ∫ yu du . T 0

(A.8)

In our case, the threshold is assumed zero, resulting in an expression equivalent to equation (4) (A.9)

C[0,T ] = a + b y .

The yield dependent term ( y − h ) in equation (A.7) is equivalent to an Asian forward contract with strike price h. Asian forward and option contracts are increasingly popular derivatives used for trading in specialized exchanges, such as those for agricultural yield or temperature. Their dependence on the underlying process yu over

A.4

the entire growing season, so-called path dependence, leads to additional valuation challenges, as discussed in the article.

Proof of Proposition [Inequality (8)] The proposition is true if, under the stated assumptions, inequality (8) holds. To show this, consider the first order condition of inequality (5) w.r.t. the IOF forward price p2 , ∂E ρ ⎡⎣ z (α ) ⎤⎦ ∂p2

(A.10)

{

}

+ + = E βT ⎡α yT + ( yT − k 1 ) − ( (1 + α ) yT − k 1 ) ⎤ , ⎣⎢ ⎦⎥

where the discount factor at harvest βT is the inverse of a bond BT with market discount rate ρ . The discount factor is a deterministic process (A.11)

= B0e − ρT .

BT

The RHS of equation (A.10) is positive if + + E ⎡⎢α yT + ( yT − k 1 ) − ( (1 + α ) yT − k 1 ) ⎤⎥ > 0 , ⎣ ⎦

(A.12)

since βT > 0 is a positive scalar. Distributing the expectation operator, + + α E [ yT ] + E ⎡⎢( yT − k 1 ) ⎤⎥ − E ⎡⎢( (1 + α ) yT − k 1 ) ⎤⎥ > 0 . ⎣ ⎦ ⎣ ⎦

(A.13)

Let α , k 1 > 0, since the relation in (A.13) is a strict equality if either α or k 1 is zero. Expressing the expectation operator as an integral, the LHS of (A.13) can be written (A.14) α



∫y

T

−∞

( x) ϕ ( x) dx +



∫ (y

T

−∞

( x) − k

)

1 +

ϕ ( x) dx −



∫ ( (1 + α ) y

T

( x) − k 1 ) ϕ ( x) dx , +

−∞

A.5

where yT ( x) = y0 e ρ +σ x , ϕ ( x) is the standard normal density, y0 is a positive constant, and

ρ = ρ − 1 2 σ 2 . Changing the limits of integration, we obtain (A.15) α



∫y

T

−∞

where x0 =





( x) ϕ ( x) dx + ∫ ( yT ( x) − k ) ϕ ( x) dx − ∫ ( (1 + α ) yT ( x) − k 1 ) ϕ ( x) dx , 1

x0

ln(k 1 / y0 ) − ρ

σ

and xα =



ln(k 1 /(1 + α ) y0 ) − ρ

σ

. Changing the limits of

integration again and canceling terms, we can rewrite (A.15) as xα

(A.16)

α ∫ yT ( x) ϕ ( x) dx − −∞

x0

∫(y

T

( x) − k 1 ) ϕ ( x) dx ≡ m (α ) .



Note that for xα ≤ x ≤ x0 , k 1 > y0 e ρ +σ x . So m (α ) > 0 , verifying the proposition.

A.6

20 18 16

combined contract revenue

14

Revenue

12 IOF contract revenue 10 8 6 4

cooperative contract revenue

2

0

o

delivery threshold y Yield

Figure 1: Combined cooperative and IOF contract revenues over a range of yield levels

10

25

combined revenue w/cheating 20

15

cooperative w/cheating

Revenue

IOF w/cheating

10

5 net cheating revenue 0 0

o

delivery threshold y Yield

Figure 2: Cheating revenues under the IOF, cooperative, and combined contracts and net cheating revenue

10

190

170

IOF price Cooperative price

150

Price per ton

.

Cooperative price w/cheating 130

110

90

70

50 1977

1979

1981

1983

1985

1987

1989

1991

1993

1995

Year

Figure 3. Investor-owned forward vs. cooperative forward prices – with and without cheating, 1977-1996

40 Delivered tons 35

Delivery volatility

30

Tons

25 20 15 10 5 0 cooperative

cooperative w/cheating

IOF

IOF w/cheating

Contract type

Figure 4. Average delivered tons and delivery volatility from 1977 to 1996, by type of forward contract

First-order condition value

4

3

2

1

0 0.4 40

0.3 38

0.2 Cheating magnitude α

36 34

0.1

32 0

30

o

Delivery threshold y

Figure 5: First-order condition values over a range of cheating magnitudes and delivery thresholds

250

Cheating option value

200

150 100

50

0 0.4 0.3

0.4 0.3

0.2 0.2 Harvest volatility σ

0.1

0.1 0

Market discount rate ρ

0

Figure 6: Cheating option values over a range of volatility levels and market discount rates

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