The impacts of crop yield and price volatility on producers’ cropping patterns: A dynamic optimal crop rotation model

Agricultural Systems 116 (2013) 52–59

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Agricultural Systems journal homepage: www.elsevier.com/locate/agsy

The impacts of crop yield and price volatility on producers’ cropping patterns: A dynamic optimal crop rotation model Ruohong Cai a,⇑, Jeffrey D. Mullen b, Michael E. Wetzstein b, John C. Bergstrom b a b

Woodrow Wilson School of Public and International Affairs, Princeton University, Princeton, NJ 08544, USA Department of Agricultural and Applied Economics, University of Georgia, Athens, GA 30602, USA

a r t i c l e

i n f o

Article history: Received 21 April 2012 Received in revised form 5 October 2012 Accepted 6 November 2012 Available online 8 December 2012 Keywords: Crop rotation Acreage response The Bellman equation

a b s t r a c t A dynamic optimization model is developed to show how crop yield and price volatility could impact acreage response under crop rotation considerations. By maximizing net present value of expected current and future farm profits, a modified Bellman equation helps optimize planting decisions. Our model is capable of simulating crop rotations with different lengths and structures. The corn–soybean rotation was simulated using the model to determine break-even prices for alternative planting decisions. Furthermore, we assume that the extent to which crop yields are penalized when skipping a rotation scheme is not fixed. Then we investigated the relationship between yield penalty levels and break-even corn price percentage changes. By considering both 1-year and 2-year carry-over effects which represent how previous crops affect current crop yield, our results indicate that producers are more likely to choose a crop rotation scheme when yield penalties are higher. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Around the world, crop rotation – planting alternative crops on the same agricultural land in consecutive seasons – has been a popular agricultural practice for centuries. Rotations are employed to reduce disease risk and pest damage while maintaining soil quality for crop growth. A prevalent example of a rotation’s agronomic benefits is the corn–soybean rotation, where soybeans provide a key nutrient (nitrogen) for corn growth (Hennessy, 2006). In terms of net returns, crop rotations generally reduce input costs and improve soil productivity, thereby increasing expected returns compared to continuous cropping (Hurd, 1994; Berzsenyi et al., 2000; Meyer-Aurich et al., 2006). They also tend to reduce yield risks (Helmers et al., 1986; Nel and Loubser, 2004; Meyer-Aurich et al., 2006). Rotational effects also impact management decisions. Mullen et al. (2005) conducted a survey focused on the determinants of Georgia producers’ crop choice and crop acreage allocation decisions. Results of the survey indicated that 80% of producers ranked rotational considerations as one of the two most important factors influencing their crop choices, and 66% of producers ranked rotational considerations as one of the two most important factors influencing their acreage allocation decisions. Although crop rotation has many benefits, it can also serve as a production constraint, ⇑ Corresponding author. Address: 411A Robertson Hall, Princeton University, Princeton, NJ 08544, USA. Tel.: +1 609 258 9897; fax: +1 609 258 6082. E-mail address: [email protected] (R. Cai). 0308-521X/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.agsy.2012.11.001

hindering the ability of producers to adjust acreage in response to changing market conditions. Switching from crop rotation to continuous cropping to take advantage of favorable market conditions such as sudden price surges could be financially beneficial in the short run while leading to long-run yield losses and associated lower net returns (Livingston et al., 2012). In recent years, as corn prices have risen sharply, many producers have allocated more acreage to corn. However, this immediate short-run return could be offset in the long run by yield losses associated with increased pest pressure and less favorable agronomic conditions due to continuous cropping. Crop rotations have been intensively studied by both agronomists and economists. The agronomic literature demonstrates that crop rotations improve or maintain crop yield while reducing input demands for fertilizers and pesticides. Johnson et al. (1998) estimated that Georgia cotton and peanut yields from a cotton–peanut rotation were 26% and 10% greater, respectively, than those from continuous cropping. In Michigan, Roberts and Swinton (1995) demonstrated that corn rotated with soybeans improved corn yields by 16% compared to continuous cropping. Vyn (2006) reported that a corn–soybean rotation in Indiana enhanced corn yields by about 6%. Discrepancies among agronomic results indicate that crop rotation effects may largely interact with various external factors such as soil type and fertilizer input, increasing the difficulty of developing economic models of crop rotation. The above agronomic model generally shows the effects of crop rotation on crop yield, which is not capable of aiding the farmers’ acreage decision under the volatility of both crop yield and crop

R. Cai et al. / Agricultural Systems 116 (2013) 52–59

price. As noted in Wu et al. (2004), there is a need for dynamic economic models of acreage decisions. Incorporating the yield effects of crop rotations is an important component of such a model. However, crop rotation effects are surprisingly omitted in many acreage response studies. Some researchers incorporate a lagged acreage variable in an econometric acreage response model, attempting to represent crop rotation effects (Bewley et al., 1987; Weersink et al., 2010). This lagged variable represents the magnitude of rotational constraints to acreage response, while the interactive effects of crop rotation on producers’ behavior are not captured. A number of economic approaches have been applied to model crop rotation. Using linear programming, a pioneer study of crop rotation was conducted by El-Nazer and McCarl (1986). The major contribution of their study is allowing the model to determine the optimal rotation, while most researchers use predetermined rotations. Multi-year crop rotations were modeled using an annual equilibrium linear programming approach. Hennessy (2006) developed an economic model of crop rotation to analyze and separate the interconnected crop rotation effects of yield-enhancement and input-saving carry-over effects. Both 1-year and multi-year carry-over effects were considered. Their model focuses on choosing among alternative rotations. Detlefsen and Jensen (2007) modeled crop rotation with network modeling. Their model provides a visual representation of the crop rotation problem. As the first in the literature to model sequential planting decisions considering crop rotations in a dynamic optimization framework, Livingston et al. (2012) used the Bellman equation, which helps solve dynamic sequential problem, to examine crop choices with price uncertainty over an infinite time horizon. Compared to previous literature, dynamic programming simulates sequential optimal decision making process and fits well into crop rotation. Various crop rotation models developed in recent years have broadly expanded our knowledge. We attempt to contribute to the literature by providing a dynamic optimization crop rotation model with minimum agronomic restrictions, such as soil type, yield response, and rotation structures. In addition, instead of focusing on fixed yield penalty taking from specific region, we conduct a sensitivity analysis towards how the variations in yield penalty related to crop rotation could affect producers’ planting decisions. As showed in the above agronomic literature, yield penalties do have large spatial variations, thus producers’ optimal decisions should vary by region as well. Livingston et al. (2012)’s yield penalty is fixed in a specific region. Although both incorporating the Bellman equation, we assume that all of the crops in a rotation are planted in the same season, while Livingston et al. (2012) assume sequential crops planted on the same land for continuous seasons. In reality, most producers actually plant all crops in crop rotation simultaneously in the same season with the purpose of reducing production risk and balancing labor load. Due to this different crop rotation planting assumption, we model dynamic optimization process differently from Livingston et al. (2012). For example, we have different state variable, control variable, and state transition function for the Bellman equation. Our model may be less computationally demanding since we only have nine states for 1-year carry-over effect, while Livingston et al. (2012)’s model contains about 2,000 evaluation points in the state space for 1-year carry-over effect. In this paper, we aim to answer the following main research question: What is the optimal cropping plan over multiple growing seasons considering the economics of crop rotation in a dynamic framework? To address this question we construct a dynamic model of economic decision making that explicitly accounts for the impact of crop rotation on economic returns over multiple growing seasons. In the remainder of the paper, we first present the modified Bellman equation for crop rotation. Then we demonstrate our key model feature – a causal flow for the state transition, which

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helps incorporate crop rotation in a dynamic programing. Finally, we take the corn–soybean rotation as an example for our crop rotation model and discuss the simulation results. 2. Methodology 2.1. Crop rotation model We start with a crop rotation system with two crops, A and B, planted on two equal-sized tracts of land. A producer plans to maximize the sum of current and expected future farm returns for certain years considering the effect of crop rotation. At the beginning of each season, the producer considers the previous season’s crop, and decides which crop to plant on the same tract of land for the current season. It is assumed that the current crop yields are determined by both the previous and current seasons’ planting decisions. If the producer decides to follow the rotation practice by switching the crops between the two tracts of land, crop yields for both A and B would be maintained at the rotational level, assuming fixed inputs of fertilizer and pesticide. If the producer decides to plant only crop A on both tracts of land, its yield in one of the tracts would decrease due to continuous cropping. Therefore, crop rotation is a Finite-Horizon Markov Decision Process, which can be simulated using the Bellman equation (Bellman, 1957). In the Bellman equation, sequential decisions are optimized to balance an immediate reward against expected future rewards (Miranda and Fackler, 2002). The basic elements for the Bellman equation, such as the state variable, control variable, and state transition function are demonstrated as follows. The producer makes planting decisions by considering the crops planted during the previous season; therefore, we take crop choice and crop yields at time t  1 as the state variable for time t.

yt1

 fy; ymg

ð1Þ

where y denotes the yield of crop y due to crop rotation, and ym denotes the reduced yield of crop y due to continuous cropping. Price is exogenous in this study and is used to convert yield into profit in the Bellman equation. Since we assume that the producer plants alternative rotational crops simultaneously during the same growing season and switches crops between two tracts of land for the next season, the size of state space is determined by the rotation length. We use A–B to represent a rotation scheme with crop A planted for 1 year and crop B planted for another year. A rotation with crops A and B could also have a different structure such as A–A–B, which means crop A is planted for two consecutive years and B is planted for the third year. For a rotation with two crops A and B, denoted by A–B, the number of elements in the state space is nine, which includes all possible combinations of yield and reduced yield for crops A and B as follows:

yt1 ðAjB; AjBM; AMjB; AjA; AjAM; AMjAM; BjB; BjBM; BMjBMÞ ð2Þ where A|B represents that crop A is planted on one tract of land with full yield, and B is planted on the other tract of land with full yield. AM or BM represents the crop with reduced yield due to continuous cropping. AM|BM is not involved in the state space. AM|BM indicates that both A and B are harvested with reduced yield due to continuous cropping, so the crops planted during the previous season must be crops A and B. While both crop A and crop B are planted for two consecutive seasons, we assume the producer will switch the tracts of land for A and B and obtain crop rotation yield A|B, as opposed to the continuous cropping yield AM|BM. Therefore, AM|BM is not included as a possible yield scenario. The control variable for the Bellman equation, the producer’s crop choice, is:

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xt fA; B; . . . ; Ng

ð3Þ

where A, B, . . . , N denotes alternative crops in a crop rotation scheme. Based on the state variable and the control variable denoted above, the modified Bellman equation for crop rotation could be written as:

V t ðyt1 Þ ¼

max fpðyt1 ; xÞ þ dV tþ1 ðgðyt1 ; xÞÞg;

xXðpðyt1 ÞÞ

t ¼ 1; 2; . . . ;

yt1 Y; ð4Þ

where Vt(yt1) is the maximum attainable sum of current and expected future farm returns, given that the system is in state yt1 in period t, x is the crop choice for the current period t, p(yt1, x) is the current season farm return, dVt+1(g(yt1, x))is expected future farm returns, and d is discount factor. The state transition function g(yt1, x) determines how the current state yt1 transits to the next state, with crop choice x. To cope with the unique characteristic of crop rotation, we develop a causal flow to visually demonstrate the state transition function for the crop rotation A–B (Fig. 1). Previous crop choice (crops A or B) and yield level (full yield or reduced yield) transits to current yield level, depending on current crop choice. To maximize the net present value of returns, the producer observes the crops planted during the previous season on two tracts of land and makes the choice between three alternative planting decisions: planting crop A on both tracts, planting crop B on both tracts, and planting crop A on one tract and crop B on the other tract. For example, in Fig. 1, the second row of the left column represents the scenario that both crops A and B were planted during the previous season, while crop A was harvested with full yield, and crop B was harvested with reduced yield. If the producer decides to plant A on both tracts during the current season, the expected current yield level will transit to the fifth row on the right column where crop A will be harvested with full yield on one tract and reduced yield on the other tract. The state transition function, the reward function, and the Bellman equation for the A–B rotation are listed in the Appendix (taking the corn–soybean rotation as an example). The above illustration of

state transition function could also be extended to represent more complicated crop rotation structures such as A–A–B and A–B–C, which are not shown in this paper. For the crop rotations with longer length, the number of elements in the state space is also larger. Compared to nine elements for the A–B rotation, the A–A–B rotation has 16 elements and the A–B–C rotation has 40 elements in their state spaces. It should be noted that the above dynamic optimization model has one strong assumption: the crop yield level at time t only depends on the crop planted at time t  1 and the planting decision at time t. However, this assumption may not be valid for some crops, for which the yield level on a particular tract of land depends on multiple lags. For example, yield at time t may depend on what was planted at both time t  1 and time t  2. To examine the impact of such a situation, we extend the model to consider 2-year carry-over effect (Fig. 2). The control variable for the new model is still the current crop choice. The state variable now reflects crop choice and yield penalty levels for the past two seasons. For the model with 1-year carry-over effect, the crops planted on the same tract of land for two consecutive seasons can be categorized into two scenarios: B ? A or A ? A, assuming crop A is planted for the current season. For the model with 2-year carryover effect, the crops planted on the same tract of land for three consecutive seasons can be categorized into four scenarios: A ? B ? A, B ? A ? A, A ? A ? A, B ? B ? A, assuming crop A is planted for the current season. A ? B ? A means that crop A was planted during the season t  2, crop B was planted during the season t  1, crop A is planted at the current season t, all on the same tract of land. To the best of our knowledge, agronomic results of yield penalty levels with 2-year carry-over effect are not generally available. We therefore make assumptions for the yield penalty levels of crop planted during current season considering rotational effects from two previous seasons. Specifically, we assume that crop A after B and B during the past two seasons has the full yield, crop A after A and B has a d% reduction in yield, crop A after B–A has a 2d% reduction in yield, crop A after A–A has a 3d% reduction in yield. For the simulation, we allow d% to vary in order to conduct

Fig. 1. A causal flow of the state transition function for the A–B rotation with 1-year carry-over effect. For each season, two crops are planted on two tracts of land denoted by cells. It should be noted that these two cells only denotes which two specific crops are planted in one season, while the left–right side orientation of these two cells is not associated with specific field. The left column denotes the current state, which is the crop choice and yield penalty level during the previous season. The right column denotes the next state depending on the current planting decision. Lines connecting the two columns denote planting decisions. Solid lines denote crop rotations, long dash lines denote growing crop A on both tracts, and dotted lines represent growing crop B on both tracts. AM or BM represents the crop with reduced yield due to continuous cropping. This figure illustrates how the state variables (crop choice and yield penalty level during the previous season) transit with the control variables (current planting decisions).

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Fig. 2. A causal flow of the state transition function for the A–B rotation with 2-year carry-over effect. For each season, two crops are planted on two tracts of land denoted by cells. It should be noted that these two cells only denotes which two specific crops are planted in one season, while the left–right side orientation of these two cells is not associated with specific field. Two columns on the left denote the current state, which is the crop choice and yield penalty level during the previous two seasons. Two columns on the right denote the next state depending on the current and last year’s planting decisions. Lines connecting the columns denote planting decisions. Solid lines denote crop rotations, long dash lines denote growing crop A on both tracts, and dotted lines denote growing crop B on both tracts. Two letters in the parenthesis in the right column denote two crops planted on the same tract of land during previous two seasons. This figure illustrates how the state variables (crop choice and yield penalty level during previous two seasons) transit with the control variables (current planting decision).

a sensitivity analysis about how producers’ decisions respond to yield penalty change. An advantage of our rotation model is that, if in the future, agronomic evidences are against our current yield penalty assumptions, we can reset penalty level to appropriate values. Compared to the 1-year carry-over effect, the number of elements in the state space for 2-year carry-over effect is larger. For the A–B rotation, there are nine elements in the state space for the model with 1-year carry-over effect only, while there are 27 elements in the state space for the model with 2-year carry-over effect (Fig. 2). Specifically, there are nine different states by crops, and each state has three yield penalty level scenarios, combining into a total of 27 elements in the state space. Nine different states by crops include (A|B, A|B), (A|A, A|B), (B|B, A|B), (A|B, A|A), (A|A, A|A), (B|B, A|A), (A|B, B|B), (A|A, B|B), and (B|B, B|B). In the parenthesis, the first element is the crops planted at time t  2, and the sec-

ond element is the crops planted at time t  1. The crop state (A|B, A|B) could come from three possible previous states: (A|B, A|B), (A|A, A|B), (B|B, A|B), and result in three yield penalty level scenarios. The same approach can be applied to extend our crop rotation model for other crop rotation structures such as A–A–B and A–B–C. In summary, for each rotation system, the number of elements in the state space for 1-year carry-over effect is the square of their possible crop combinations that could be planted on two tracts of land during the same season, for 2-year carry-over effect, it is the cubic of their possible crop combinations. A–B has three crop combinations in two tracts: A|B, A|A, and B|B; the A–A–B rotation has four combinations in three tracts: A|A|B, B|B|A, A|A|A, and B|B|B. A–B–C has 10 combinations in three tracts: A|B|C, A|A|B, B|B|A, A|A|C, C|C|A, B|B|C, C|C|B, A|A|A, B|B|B, and C|C|C. Compared to 27 elements for the A–B rotation, the A–A–B rotation has 64 elements and the A–B–C rotation has 1000 elements in their state

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spaces. Their transition functions, reward functions, and modified Bellman equation are not demonstrated in this paper. Overall, we develop crop rotation models under two assumptions: 1-year carry-over effect and 2-year carry-over effect. The model could be further extended to consider 3-year carry-over effect or even longer length. 2.2. Crop rotation simulation MATLAB utilizes the CompEcon toolbox to solve discrete time/ discrete variable dynamic programming problem (Fackler, 2010), and is used for our model simulation. Given the terminal value of Vt+1(g(yt1, x)), the decision is solved recursively by repeated application of the Bellman equation. MATLAB compares the value of Vt+1(g(yt1, x)) for each time t, and provides the optimal decision for each period. The value for each Vt+1(g(yt1, x)) includes current and discounted future rewards. The current reward for each period is a producers’ immediate profit:

pitðdÞ ¼

N X

ðPit Y itðdÞ  C it Þ

ð5Þ

i¼1

The above profit function is the profit summation for the crops on both tracts of land under planting decision d. Take a corn– soybean rotation as an example. If keeping a rotation scheme is decided, the current expected profit will be the total of corn profit and soybeans profit, both with rotational yields. If growing all corn is decided after a previous rotation scheme, the current expected profit will be corn profit with rotational yields and corn profit with reduced yields. If planting all soybeans is decided after a previous rotation scheme, the current expected profit will be soybean profit with rotational yields and soybeans profit with reduced yields. MATLAB compares three profit bundles with each adding their expected future rewards. The transition function is incorporated in the MATLAB program by using causal flows in Figs. 1 and 2 as algorithm.

maximizer and price-taker. It is further assumed that the producer owns two equally sized tracts of land. At the beginning of each season, the producer decides which crop to plant on each tract based on yield and price expectations. For 2-year carry-over effect, we assume corn after soybean and soybean planted during previous two seasons has the full yield, corn after corn and soybean has a d% reduction in yield, corn after soybean and corn has a 2d% reduction in yield, and corn after corn and corn has a 3d% reduction in yield. In addition, we set d% to be 5%. A different yield penalty could be assumed for soybeans. For example, soybean after soybean and corn has a 4% reduction in yield instead of 5% as the case for corn. For simplicity, we assume that yield penalties for corn and soybeans are the same in a corn– soybean rotation. We also check the consistency between 1-year and 2-year carry-over effects models. In order to do that, we assume corn after soybean and soybean planted during the previous two seasons has the full yield, corn after corn and soybean has the full yield, corn after soybean and corn has a d% reduction in yield, and corn after corn and corn has a d% reduction in yield. This assumption reduces the model with 2-year carry-over effect to the case with 1-year carry-over effect. The results for the 2-year carry-over effect model which are reduced to the 1-year carry-over effect model do repeat the results for the 1-year carry-over effect model. In this paper, we specify four planting scenarios for producer’s response. All-corn scenario denotes that the producer only plant corn for each of the next 5 years, all-soybeans scenario denotes that the producer only plant soybeans for each of the next 5 years, all-rotation scenario means that the producer plant both corn and soybeans for each of the next 5 years. Mixed scenario denotes that the producer plants one crop in some years, and plants two crops in other years (mixed by continuous and rotational practices). Although the producer’s planting decisions are optimized for each year, we do not discuss the order of these planting decisions for simplicity.

2.3. Data and model calibration

3. Results and discussion

The model was simulated on the corn–soybean rotation, with both the 1-year and 2-year carry-over effects. We first simulate producers’ planting decisions over next 5 years based on current USDA projection of crop yields and prices, with fixed yield penalty level (Westcott, 2011). We then investigate the break-even price percentage change, which represents how much price percentage should change in order for producers to switch between continuous and rotational cropping for all 5 years. Corn price is allowed to vary based on USDA 5-year price projection. For example, if corn price is 5% higher for each of next 5 years compared to USDA projection, we say corn price is at 105% price level. Furthermore, we relax the yield penalty level to conduct a sensitivity analysis of producers’ decisions. Yield penalty level is allowed to vary based on rotation yield. For example, a 5% of yield penalty means yield decreases by 5% from full yield rotation scheme. For 1-year carry-over effect, the fixed yield penalty of the corn– soybean rotation is retrieved from a compilation of known published data comparing continuous cropping to a corn–soybean rotation (Erickson, 2008). The data show that corn yield in continuous cropping is on average 7.8% lower than the corn yield in rotation. The soybeans yield after continuous cropping is 14.5% lower than the soybeans rotation yield. Since many producers use crop rotation systems, we assume that USDA yield projections are equal to the rotational yields (USDA, 2011). Crop yields under continuous cropping are reduced based on yield penalty level retrieved from Erickson (2008). Input and output prices are also retrieved from USDA projections. The producer is assumed to be a profit-

3.1. One-year carry-over effect Based on current USDA projections for the next 5 years, the simulation result indicates that the producer will plant corn on both tracts for each of next 5 years – denoted as all-corn scenario. As long as corn prices are higher than 98% of the USDA corn price projections, the producer plant corn for all next 5 years. The upper and lower bounds of break-even corn price percentage changes for crop rotations for all 5 years are 88% and 84%, respectively. If corn prices decrease to less than 70% of the projections level, the producer will grow soybeans for all 5 years. For corn price percentage changes lower than 98% but higher than 88%, or lower than 84% but higher than 70%, the planting decisions follow a mixed scenario. Furthermore, we allow yield penalty levels to vary to show the response of break-even corn price percentage changes (Fig. 3). It is observed that the lower the yield penalty levels, the less likely a producer is to stay in rotation for all 5 years, since rotational benefits are relatively small. When yield penalty levels are higher, producers are more likely to stay in rotation for all 5 years since rotational benefit are relatively large. This result is consistent with what Livingston et al. (2012) found, that the agronomic benefits of rotations leads to a much more inelastic response of planting decisions to price change, and rotating is extremely close to optimal. We reconfirm their conclusion by not only relaxing fixed price, but also fixed yield and fixed yield penalty. In our result, when the yield penalty level is close to 0%, all-rotation is not possible no matter how corn price varies. This is because crop rotation

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Fig. 3. The response of break-even corn price to yield penalty considering 1-year carry-over effect. It demonstrates how different yield penalties alter the range of the ratio of corn prices to USDA baseline prices for producer to select certain planting schemes. All-corn area means producer plants continuous corn over a 5-year horizon, all-soybeans area means producer plants continuous soybeans over a 5-year horizon, all-rotation area means producer keeps the corn–soybean rotation scheme over a 5-year horizon, and mixed area means both continuous and rotational scheme exists over a 5-year horizon. This analysis assumes no change of USDA projection of soybeans prices.

yields are similar to continuous cropping yields when the yield penalty level is close to 0%. Once the yield penalty level goes beyond 6.3%, all-rotation scenario becomes possible. For higher yield penalty levels, break-even corn price percentage changes are also larger which indicates that corn price needs to be higher for producers to plant corn for all 5 years, and corn price needs to be lower for producers to plant soybeans for all 5 years. This is because a higher yield penalty makes continuous cropping less profitable and therefore large price differences between corn and soybeans prices are required for producers to switch to continuous cropping. It is interesting to find out that if we set corn price to be zero, as long as penalty level are below about 77%, producers will plant all soybeans for the next 5 years. However, if we let penalty level increases beyond 77%, a mixed scenario emerges, which means producers plant corn in at least one of the next 5 years, even though corn prices are zero. This could happen when soybeans price is relatively high for the next season, considering the high penalty level (more than 77%), producer would rather sacrifice this

year’s return to take advantage of next season’s high soybeans price. This is an extreme example, and it is unlikely for crops to have a zero price, but in reality, producers do include cover crop in a rotation, which is not planted directly for profit, but providing better nutrition for the other crop. 3.2. Two-year carry-over effect Next, we simulate a corn–soybean rotation with 2-year carryover effect. Simulation results indicate that, if USDA corn price projections are higher than 108% of the current level, the producer will follow all-corn scenario. If corn prices decrease by over 30%, the producer will follow all-soybeans. No level of corn price change can be found for an all-rotation scenario which is different from the 1-year carry-over effect. We then investigate the response of break-even corn price percentage changes to varied yield penalty levels (Fig. 4), allowing yield penalty to vary instead of staying at 5% level. Similar to the

Fig. 4. The response of break-even corn price to yield penalty change considering 2-year carry-over effect. It demonstrates how different yield penalties alter the range of the ratio of corn prices to USDA baseline prices for producer to select certain planting schemes. The highest yield penalty level for 2-year carry-over effect is 33.3%. All-corn area means producer plants continuous corn over a 5-year horizon, all-soybeans area means producer plants continuous soybeans over a 5-year horizon, and mixed area means both continuous and rotational scheme exists over a 5-year horizon. This analysis assumes no change of USDA projection of soybeans prices.

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results from 1-year carry-over effect, we observe that the producer is more likely to practice continuous cropping when the penalty level is lower. Also, when the penalty level increases, corn price needs to be higher for producers to choose continuous cropping. The simulation of 2-year carry-over effect also shows different results compared to the one with 1-year carry-over effect. At the same yield penalty level, break-even corn price percentage change for all-corn scenario are higher than that in the 1-year carry-over effect case. This indicates that 2-year carry-over effect enforces larger rotational constraint, and the producer is more likely to avoid continuous cropping. Another difference is that all-rotation scenario is not possible in the range of yield penalty levels investigated when 2-year carry-over effect is assumed, indicating that 2-year carry-over effect create a more complex system. Producers are more likely to mix both crop rotation and continuous cropping schemes over a 5-year period in order to maximize the net present value of farm returns. 4. Conclusions In this paper, a dynamic optimization model incorporating the Bellman equation was developed to simulate acreage responses considering crop rotation effect. A key feature of our model is using a causal flow to illustrate the state transit function of the Bellman equation. The crop rotation models were developed for both 1-year and 2-year carry-over effects assumptions. The simulation results indicate that, by assuming 1-year carry-over effect, continuous corn cropping is the most profitable choice based on current USDA projections of crop yields and prices for the next 5 years. Assuming 2-year carry-over effect, the optimal choice involves both continuous cropping and crop rotation within a 5-year period based on the latest USDA projections. Furthermore, when allowing yield penalty level to vary, we observe that higher penalty levels increases break-even corn prices associated with all-corn scenario (or decreases break-even corn prices associated with all-soybeans scenario). We also find that all-rotation scenario is possible when the yield penalty increases to a certain level under 1-year carryover effect assumption, while it is not possible in the range of yield penalty levels or corn prices investigated under 2-year carry-over effect assumption. As one of the initial attempts to apply the Bellman equation to acreage response considering crop rotation, our model has

some major assumptions. External inputs were fixed for simplicity. Future research could improve this model by including interactions between crop yield and fertilizer usage (Mallarino et al., 2005). In our model, producers’ acreage responses will not cause dynamic price responses since they are assumed to be price takers, while in the real world case, large producers’ acreage responses could drive price change. Another limitation for the study is the assumption of four yield penalty levels for 2-year carry-over effect due to the lack of agronomic data. Future related research should incorporate more realistic yield penalty levels when related agronomic data becomes available. Also, it should be noted that crop choice could be affected by labor or capital constraints which keep producer from an optimal crop choice. Despite the above limitations, our model simulates the corn– soybean rotation and demonstrates the potential magnitude of rotational effect in acreage responses. The results of this model are expected to pave the way for future research in economic modeling of crop rotation in a dynamic simulation framework. The model is designed to be applicable to crop rotations with minimum agronomic restrictions. It is also assumed that each crop of a rotation system is grown simultaneously in the same season to simulate the real-world cases. Therefore, our results provide valuable information to both agricultural producers and policymakers. For example, agricultural producers could see the importance of crop rotation from our simulation results and consider more in long term profit optimization. Policy makers should adjust agricultural policies to encourage crop rotation practices, such as promoting education about crop rotation knowledge or setting up agricultural subsidy programs for crop rotation. Appendix A A.1. The reward functions

The state v ariable Y fCjS; CjSM; CMjS; CjC; CjCM; CMjCM; SjS; SjSM; SMjSMg; t f1; 2; 3; . . . ; ng The action v ariable xfcroprotation; all corn; all soybeansg

8 if Y t1 ¼ ðCjS; or CjSM; or CMjSÞ and x ¼ rotation; > < ðP c  C c ÞY c þ ðPs  C s ÞY s ; f ðY t1 ; xÞ ¼ ðP c  C c ÞY c þ ðPc  C c ÞY cm ; if Y t1 ¼ ðCjS; or CjSM; or CMjSÞ and x ¼ corn; > : ðP s  C s ÞY s þ ðPs  C s ÞY sm ; if Y t1 ¼ ðCjS; or CjSM; or CMjSÞ and x ¼ soybeans; 8 if Y t1 ¼ ðCjC; or CjCM; or CMjCMÞ and x ¼ rotation; > < ðP c  C c ÞY cm þ ðPs  C s ÞY s ; f ðY t1 ; xÞ ¼ ðP c  C c ÞY cm þ ðPc  C c ÞY cm ; if Y t1 ¼ ðCjC; or CjCM; or CMjCMÞ and x ¼ corn; > : ðP s  C s ÞY s þ ðPs  C s ÞY s ; if Y t1 ¼ ðCjC; or CjCM; or CMjCMÞ and x ¼ soybeans; 8 > < ðP c  C c ÞY c þ ðPs  C s ÞY s m; if Y t1 ¼ ðSjS; or SjSM; or SMjSMÞ and x ¼ rotation; f ðY t1 ; xÞ ¼ ðP c  C c ÞY c þ ðPc  C c ÞY c ; if Y t1 ¼ ðSjS; or SjSM; or SMjSMÞ and x ¼ corn; > : ðP s  C s ÞY sm þ ðPs  C s ÞY sm ; if Y t1 ¼ ðSjS; or SjSM; or SMjSMÞ and x ¼ soybeans;

R. Cai et al. / Agricultural Systems 116 (2013) 52–59

A.2. The transition functions

8 < 9t þ 1; if Y t ¼ ðCjS; or CjSM; or CMjSÞ and x ¼ rotation; gðY t ; xÞ ¼ 9t þ 5; if Y t ¼ ðCjS; or CjSM; or CMjSÞ and x ¼ corn; : 9t þ 8; if Y t ¼ ðCjS; or CjSM; or CMjSÞ and x ¼ soybeans; 8 > < 9t þ 3; if Y t ¼ ðCjC; or CjCM; or CMjCMÞ and x ¼ rotation; gðY t ;xÞ ¼ 9t þ 6; if Y t ¼ ðCjC; or CjCM; or CMjCMÞ and x ¼ corn; > : 9t þ 7; if Y t ¼ ðCjC; or CjCM; or CMjCMÞ and x ¼ soybeans; 8 > < 9t þ 3; if Y t ¼ ðSjS; or SjSM; or SMjSMÞ and x ¼ rotation; gðY t ; xÞ ¼ 9t þ 4; if Y t ¼ ðSjS; or SjSM; or SMjSMÞ and x ¼ corn; > : 9t þ 9; if Y t ¼ ðSjS; or SjSM; or SMjSMÞ and x ¼ soybeans;

A.3. The Bellman equation

VðY t1 Þ ¼ maxfðPc  C c ÞY c þ ðPs  C s ÞY s þ dVð9t þ 1Þ; ðPc  C c ÞY c þ ðPc  C c ÞY cm þ dVð9t þ 5Þ; ðPs  C s ÞY s þ ðPs  C s ÞY sm þ dVð9t þ 8Þg; if Y t1 ¼ ðCjS; or CjSM; or CMjSÞ VðY t1 Þ ¼ maxfðPc  C c ÞY cm þ ðPs  C s ÞY s þ dVð9t þ 1Þ; ðPc  C c ÞY cm þ ðPc  C c ÞY cm þ dVð9t þ 5Þ; ðPs  C s ÞY s þ ðPs  C s ÞY s þ dVð9t þ 8Þg; if

Y t1

¼ ðCjC; or CjCM; or CMjCMÞ VðY t1 Þ ¼ maxfðPc  C c ÞY c þ ðPs  C s ÞY sm þ dVð9t þ 1Þ;  C c ÞY c þ ðP c  C c ÞY c þ dVð9t þ 5Þ;  C s ÞY sm þ dVð9t þ 8Þg;

ðPc

ðPs  C s ÞY sm þ ðPs

if Y t1

¼ ðSjS; or SjSM; or SMjSMÞ Note: C denotes corn, S denotes soybeans, CM denotes corn with reduced yield, SM denotes soybeans with reduced yield. References Bellman, R.E., 1957. Dynamic Programming. Princeton University Press, Princeton NJ.

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