Private grower optimization

Consider N identical growers who use one type of plastic mulch in growing a single crop on a homogeneous farm under an infinite planning horizon. Initially each representative grower has perfect information of the degradation rate of mulches, and each grower is indexed by i. We assume that the market can offer different types of plastic mulches characterized by an in-soil degradation rate, ${\delta _{i,t}}$ where the degradation rate parameter ${\delta _{i,t}}$ is assumed to be between 0% and 100% ( $0{ \le _{i,t}} \le 1$ ).Footnote ⁴ The degradation rate equal to 0% represents the PEMs that have no degradation during a crop year t. If the in-soil degradation rate is greater than 0% but no more than 100% (0 < ${\delta _{i,t}} \le $ 1), the plastic mulch is identified as a BDM. When the degradation rate is equal to 100%, it implies the completely degradable BDMs. Because of environmental differences for degradation rates in-soil as opposed to above the soil, the above-soil degradation rate is denoted as ${\xi _{i,t}}{\rm{\;}} = \theta _{i,t}^{ - 1}({\delta _{i,t}})$ . Then the function ${\delta _{i,t}} = \theta \left( {{\xi _{i,t}}} \right)$ expresses the in-soil degradation rate as a function of above-soil degradation rate. We assume the above-soil degradation rate is a monotone function of ${\delta _{i,t}}$ , and the value of $\;{\xi _{i,t}},{\rm{\;}} = $ $\theta _{i,t}^{ - 1}\left( . \right)$ is between 0% and 100% as well.Footnote ⁵ No matter the type of plastic mulches used by the farmer, the waste management decision on disposal of used plastic mulches will affect the plastic stock of pollution in the farmland soil. Henceforth, to focus the analysis, we assume if the farmer does not dispose of the plastic mulch waste after the growing season, then the used plastic mulches will be incorporated in the soil contributing to the plastic pollutant stock in the farmland soil, ${S_{i,t}}$ .Footnote ⁶

We assume the production function for each grower, i, $Q\left( {{\xi _{i,t}};{S_{i,t}}} \right) = Q\left( {{\theta ^{ - 1}}\left( {{\delta _{it}}} \right);{S_{i,t}}} \right),$ is a continuously differentiable strictly decreasing function of the degradation rate, ${\delta _{it}}$ and the plastic pollutant stock in the farmland soil, ${S_{i,t}}$ , ( ${Q_\delta }$ <0, $\;{Q_S}$ <0).Footnote ⁷ The production function $Q\left( . \right)$ is strictly concave ( ${Q_{\delta \delta }} \lt 0,\;{Q_{SS}} \lt 0,\;{Q_{S\delta }} = 0,$ ). The grower i is assumed to sell all the crops at the market price, ${P_t}$ . Hence, ${P_t}Q\left( {{\theta ^{ - 1}}\left( {{\delta _{it}}} \right);{S_{i,t}}} \right)$ represents the revenue of grower i’s farm at the end of the crop year. Figure 1 illustrates a grower’s decision flowchart, and box A is corresponding to the revenue.

Figure 1. Grower decision-making flow chartFootnote ⁹ .

The cost function has two components, the production costs and the disposal costs. The grower’s private input cost function, $C\left( {{\delta _{i,t}}} \right)$ , is a strictly increasing function of the degradation rate ( ${C_\delta } \gt 0$ ) to recognize that the cost of BDMs is higher than that of PEMs. The input cost function is non-concave ( ${C_{\delta \delta }} \ge 0).\;$ The production costs are represented by box B in Figure 1. The term $w{h_t}\left( {\theta _{it}^{ - 1}\left( {{\delta _{it}}} \right),{z_{it}}} \right)$ represents the plastic-mulch-waste disposal cost, where $w$ (measured in $/lb) is the disposal rate.Footnote ⁸ ${h_t}\left( {\theta _{it}^{ - 1}\left( {{\delta _{it}}} \right),{z_{it}}} \right)$ is the amount of plastic mulch waste disposed at disposal facilities (measured in pounds), where ${h_t}\left( . \right)$ is a non-decreasing function of the portion of plastic mulch waste, ${z_{it}}$ , handled by the disposal facility ( ${h_z} \ge 0)$ , but a strictly decreasing function of the degradation rate ( ${h_\delta } \lt 0$ ). It is non-concave ( ${h_{\delta \delta }} \ge 0)\;$ in the degradation rate. The disposal costs are represented by box C in Figure 1.

Following previous literature, the discount factor $0 \lt \rho \lt 1\;$ indicates that the grower values the current profit more than the future profit (Conrad and Olson Reference Conrad and Olson1992; Martínez and Albiac Reference Martínez and Albiac2004). Therefore, the grower’s objective of maximizing the discounted sum of current and future annual profits from selling a single crop is specified as

$$Max\;_\left\{ {{\delta _{it}},{z_{it}}} \right\}\;\pi \left( {{\delta _{it}},{z_{it}};{S_{it}},w} \right)$$

(1) $$= Max\;_\left\{ {{\delta _{it}},{z_{it}}} \right\}\;\mathop \sum \limits_{t = 0}^\infty {\rho ^t}\left[ {{P_t}{Q_{it}}\left( {\theta _{it}^{ - 1}\left( {{\delta _{it}}} \right);{S_{it}}} \right) - {C_{it}}\left( {{\delta _{it}}} \right) - {w}{h_t}\left( {\theta _{it}^{ - 1}\left( {{\delta _{it}}} \right),{z_{it}}} \right)} \right]$$

Subject to

(2) $${S_{it + 1}} = {G_{it}}\left( {{\delta _{it}},{z_{it}};{S_{it}},w,\overline {{S_{base,\;t}}} } \right)$$

(3) $$0 \le {\delta _{it}} \le 1$$

(4) $${z_{it}} = 0\;or\;1$$

(5) $${S_0} = \bar S\;and\;{h_t} \ge 0\;$$

where the grower i simultaneously chooses a mulch with degradation rate, ${\delta _{it}}$ for the local environment, and the portion of plastic mulch waste handled by the disposal facility, ${z_{it}}$ , which can have a value between 0 and 1. In practice most growers either dispose of all ( ${z_{it}} = 1$ ) or none ( ${z_{it}} = 0$ ) of the plastic mulch waste, which form individual corner solutions.Footnote ¹⁰

Equation 2 is the function of the evolution for the plastic pollutant stock in the farm soil. At the end of each crop year, the plastic pollutant stock in the farm soil, ${S_{i,t + 1}}$ , will come from two sources: the plastic pollutant stock in the farmland soil from previous crop years and the in-flow of undisposed plastic mulch residue from the current crop year. At the beginning of the crop year, the farmland soil has contained plastic pollutant stock, ${S_{it}}$ , but during the crop year, ${S_{it}}$ degrades in the soil with a degradation rate, ${\delta _{it}}$ . Therefore, ${G_{it}}\left( . \right)$ is an increasing function of the plastic pollutant stock, ${S_{it}}\left( {0 \le {G_s} \le 1} \right),$ but a decreasing function of the in-soil degradation rate, ${\delta _{it}}$ . $\overline {{S_{base,\;t}}} $ is a base level of residue from PEM (or BDM for that matter) whether the mulch goes to a landfill or stays on the property. We also assume that if the completely degradable BDMs are used, the accumulated plastic mulch pollutant stock will equal to a base level, $\overline {{S_{base,\;t}}} $ . We assume ${G_{it}}\left( . \right)$ is subject to the curvature conditions: ${G_{\delta \delta }} \ge 0,{G_{ss}} \ge 0\;and\;{G_{s\delta }} \le 0$ .

The discrete current value Hamiltonian (Shone Reference Shone2003) is

(6) $$H = {P_t}{Q_{it}}\left( . \right) - {C_{it}}\left( . \right) - w{h_t}\left( . \right) + \rho {\lambda _{i,t + 1}}\left( {{G_{it}}\left( . \right) - {S_{it}}} \right)$$

where ${\lambda _{i,t + 1}}$ can be interpreted as the shadow price whose value is negative. The shadow price measures the loss in future profit due to additional plastic pollutant stock in the farmland soil reducing the soil productivity. The Karush-Kuhn-Tucker conditions (equations 7 - 10) and the sufficient transversality condition (equation 11) are as follows:

$${{\partial H} \over {\partial \delta }} = {P_t}_\delta - {C_\delta } - w{h_\delta } + \rho {\lambda _{i,t + 1}}{G_\delta } \gt 0\;\;yields \Rightarrow \;\;\delta _{it}^* = 1\;$$

(7) $${{\partial H} \over {\partial \delta }} = {P_t}{Q_\delta } - {C_\delta } - w{h_\delta } + \rho {\lambda _{i,t + 1}}{G_\delta } \lt 0\;\;yields \Rightarrow \;\delta _{it}^* = 0\;$$

$${{\partial H} \over {\partial \delta }} = {P_t}{Q_\delta } - {C_\delta } - w{h_\delta } + \rho {\lambda _{i,t + 1}}{G_\delta } = 0\;\;yields \Rightarrow \;0 \lt \delta _{it}^* \lt 1$$

(8) $${{\partial H} \over {\partial z}} = - w{h_z} + \rho {\lambda _{i,t + 1}}{G_z} \lt 0\;\;yields \Rightarrow \;\;z_{it}^* = 0$$

$${{\partial H} \over {\partial z}} = - {h_z} + \rho {\lambda _{i,t + 1}}{G_z} \gt 0\;\;yields \Rightarrow \;z_{it}^* = 1$$

Footnote ¹¹

(9) $${\partial \over {\partial S}} = {P_t}{Q_S} + \rho {\lambda _{i,t + 1}}\left( {{G_S} - 1} \right) = - \rho {\lambda _{i,t + 1}} + {\lambda _{i,t}}$$

(10) $${{\partial H} \over {\partial \rho {\lambda _{i,t + 1}}}} = {G_{it}}\left( {{\delta _{it}},{z_{it}};{S_{it}},w,\overline {{S_{base,\;t}}} } \right) - {S_{it}} \ge 0$$

(11) $${\rho ^t}{\lambda _{i,t}}{S_{i,t}} = 0\;$$

We use ${\lambda ^*},\;{\delta ^*},{S^*},$ and ${z^*}$ to denote the steady-state solutions. Condition (8) quantifies the differences between the marginal cost ( $ - w{h_z}$ ) and marginal benefit ( $ - \rho {\lambda _{i,t + 1}}{G_z}$ ) associated with the disposal decision. Both the cost and benefit of disposal are not associated with crop production function or crop price. If the marginal cost is large than the marginal benefit of disposing mulch waste, i.e., $ - w{h_z} + \rho {\lambda _{i,t + 1}}{G_z} \lt 0$ , the grower will till all used plastic mulches into the farmland soil. This practice leads to a continuous accumulation of plastic pollutants, diminishing soil productivity and ultimately causing the grower to cease farming operations. Consequently, the model will converge towards the transversality condition (11). An increase in disposal costs and fees, $w$ , will elevate the marginal cost of disposing mulch waste, making condition $ - w{h_z} + \rho {\lambda _{i,t + 1}}{G_z} \lt 0$ more likely to be satisfied. Therefore, the grower is more inclined to till all used plastic mulches into the soil.

However, if the marginal cost is smaller than the marginal benefit of disposing mulch waste, i.e., $\; - w{h_z} + \rho {\lambda _{i,t + 1}}{G_z} \gt 0$ the grower will opt to send all used plastic mulches to a disposal facility. In this case, the plastic pollutant stock in the farmland soil is determined by an accumulated base level of residue from mulch even if it goes to a disposal facility.

Considering condition (7) that identifies the degradation rate decision, we first discuss the boundary solutions. Suppose that ${P_t}{Q_\delta } - {C_\delta } - w{h_\delta } + \rho {\lambda _{i,t + 1}}{G_\delta } \gt 0,$ then the optimal in-soil degradation rate ${\delta ^*} = 1$ (the fully degradable BDMs). Therefore, if $z_{it}^*$ is equal to 0, the optimal pollutant stock in the farmland soil ${S^*}$ will only be determined by initial plastic residue in the farmland soil. Considering the case in which ${P_t}{Q_\delta } - {C_\delta } - w{h_\delta } + \rho {\lambda _{i,t + 1}}{G_\delta } \lt 0,$ we obtain that the optimal in-soil degradation rate ${\delta ^*} = 0$ , which is the conventional PEMs.

We now turn to the interior solution, where $0 \lt \delta _{it}^* \lt 1$ . We follow Schnitkey and Miranda (Reference Schnitkey and Miranda1993) to drop the time subscript t, and grower subscript i, and denote the steady state for a representative grower as ${\lambda ^*}$ , ${\delta ^*},{S^*},{z^*}$ . Therefore, the optimal degradation rate at the steady state satisfies the equation (12) (see derivation is in Appendix A)

(12) $${\delta ^*} = {\Psi ^{ - 1}}\left( {{{{C_\delta } + {\rm{w}}{h_\delta }} \over {\bar P}}} \right)$$

where ${\Psi ^{ - 1}}\left( . \right)$ is the inverse function of $\Psi \left( . \right)$ with respect to the degradation rate, and it is a strictly decreasing function of ${{({C_\delta } + w{h_\delta })} \over {\bar P}}$ . Equation (12) identifies the steady-state degradation responses relative to changes in key parameters. Consequently, when the crop price, $\bar P,$ increases, the grower is more likely to use plastic mulches with a higher degradation rate. In other words, when crop prices are higher, farmers have the financial incentive to prioritize long-term soil improvement by choosing the faster-degrading mulch with higher upfront cost. Equation (12) also predicts that for a higher the disposal rate and fee, $\;w$ , then a mulch with a higher degradation rate will be chosen by the grower.Footnote ¹² If disposal fees for plastic mulch are very high in the farmer’s area, the economic benefits of faster degradation could outweigh other considerations. A higher degradation rate means less plastic to haul away and pay for at the end of the season. Essentially, the analysis of equation (12) reveals that higher crop prices or higher disposal cost incentivize growers to invest in practices that maximize long-term yield and minimize costs. Growers will weigh the benefits of faster degradation (reduced disposal costs and soil health) against the potentially higher upfront cost of BDMs. The relationship revealed in this research aligns with existing research on the private interest of farmers in maintaining or increasing soil quality. Note that Issanchou et al. (Reference Issanchou, Daniel, Dupraz and Ropars-Collet2019) reported that crop prices positively influence farmer investments in soil conservation.

Biodegradation standards

In this section, we relax the assumption of perfect information regarding degradation rate. Here, growers rely on biodegradation standards and certifications to identify biodegradable plastic mulches. Plastic producers substantiate degradability claims by adhering to national or international standards (Federal Trade Commission 2012, 2014). Several standards guide the identification of biodegradable plastics in various environments (compost, marine, etc.): ASTM D6400 and D5526-11 (US), EN 13432 and EN 17033 (Europe/Japan), and ISO 17088. These standards establish protocols, materials, and conditions for testing, as well as define degradation rate benchmarks for certified soil-biodegradable plastics. For instance, ASTM D6400 certifies a plastic as soil-biodegradable if at least 90% of its organic carbon converts to CO₂ within 180 days under controlled laboratory conditions (Avérous and Halley Reference Avérous and Halley2014).

In other words, asymmetric information creates a challenge for BDMs production. When farmers lack complete knowledge of BDMs’ degradation rate, BDMs manufactures are required to produce BDMs meeting the standard, the variety of plastic mulches offered with varying degradation rates may be lower compared to a scenario with perfect information. To reflect such conditions, the constraint (3) will be modified as follows:

(13) $${\delta _{it}} = 0\;\;or\;\;0.9 \le {\delta _{it}} \le 1$$

Let ${\delta ^C}\;$ denote the optimal degradation rate under the current biodegradation certification scheme. If ${\delta ^*}$ defined in equation (13) is no smaller than 90% ( ${\delta ^*} \ge 0.9$ ), ${\delta ^C} = {\delta ^*}$ . However, if ${\delta ^*} \lt 0.9$ , then the optimal choice will depend on the general shape of a grower’s profit function.

Figure 2 depicts three plausible scenarios. Figure 2(a) depicts the scenario that the lifetime profit is larger when a grower chooses PEMs rather than BDMs meeting standards. A grower would gain a higher lifetime profit by using BDMs meeting standards with a 90% degradation rate than PEMs if the scenario represented in Figure 2(b) happened. We also can observe the situation depicted in Figure 2(c) where there is no difference between using PEMs or BDMs meeting standards with a 90% degradation rate to achieve the highest lifetime profit. Simulation methods are applied in section “Empirical model and data” to characterize the shape of a grower’s profit function so that we are able to study the grower’s optimal choice under current biodegradation standards in section “Results and discussion.”

Figure 2. Lifetime profits comparison between using PEMs and using BDMs meeting standards.a). PEMs is optimum b). BDMs meeting standards is optimum c). Both PEMs and BDMs meeting standards are optimum. Note: both point A and B achieve a local optimum. To obtain the global optimum, we need to compare the profit at point A and B, and the one having higher profit will be the global optimum.

Social planner optimization

This section expands our model to consider the social optimum and the policy implications of aligning individual grower decisions with the social planner’s choice (assuming perfect information).

While growers manage plastic mulch waste on-farm to minimize reductions in crop productivity from soil pollution, negative environmental externalities still persist. Landfilling, a common disposal method, contributes to the global plastic waste crisis, harming terrestrial (Dreyer et al Reference Dreyer, Fourie and Kok1999; Lgbokwe et al Reference Lgbokwe, Kolo and Egwu2003; Omidi et al Reference Omidi, Naeemipoor and Hosseini2012) and marine animals (Cameron and Dudek Reference Cameron and Dudek2009; Derraik Reference Derraik2002; Joyner and Frew Reference Joyner and Frew1991). Incineration releases greenhouse gases by converting petroleum-based carbon into atmospheric carbon (Ellis et al Reference Ellis, Kantner, Saab and Watson2005; Gautam Reference Gautam2009).

To mitigate negative externalities arising from pollution, social planners can employ various environmental policies. Three primary instruments exist for pollution control, aiming to achieve a socially optimal outcome (Baumol and Oates Reference Baumol and Oates1988; Kolstad Reference Kolstad2011). Command-and-control regulations directly limit pollution emissions or mandate specific technologies. While offering certainty in pollution reduction and potentially simplifying complex environmental processes, this approach can discourage innovation in pollution control methods (Kolstad Reference Kolstad2011). Pollution fees (or taxes) incentivize polluters to internalize external costs by imposing a charge on their emissions (Pigou Reference Pigou1932). Pollution permits utilize a market-based approach where a limited number of permits are issued and traded among firms. Both fees and permits are price-based incentives that require less information than command-and-control and can be cost-effective due to market-driven adjustments. However, these price-based instruments may struggle to address the complexities of environmental systems without becoming overly complicated themselves (Kolstad Reference Kolstad2011) and may face challenges in accurately measuring marginal social damage (Baumol and Oates Reference Baumol, Oates, Bohm and Kneese1971).

Consequently, many scholars advocate for a hybrid approach that combines the strengths of both quantity-based and price-based instruments (Baumol and Oates Reference Baumol, Oates, Bohm and Kneese1971; Jacobs and Timmons Reference Jacobs and Timmons1974; Moffitt et al. Reference Moffitt, Zilberman and Just1978; Schnitkey and Miranda Reference Schnitkey and Miranda1993; Holland Reference Holland2012). This typically involves setting a cap on total pollution allowed and using a market mechanism (such as a tax or a tradable permit system with a price ceiling) to ensure the cap is not exceeded. This approach allows policymakers to achieve emission reduction targets while encouraging innovation in pollution control.

Therefore, we adopt the hybrid approach in this research and assume that the social planner aims to control both plastic pollutant stock in farmland and waste disposed of in landfill facilities with a limitation. Such a limitation can be expressed as an extra constraint for the optimization problem

(14) $$\mathop \sum \limits_{i = 1}^N {S_{it}} + \mathop \sum \limits_{i = 1}^N {h_{it}} \le \overline {{D_t}} $$

where the maximum level of the plastic mulch waste is exogenously determined by the social planner as $\overline {{D_t}} $ .

The socially optimal problem is to maximize the summation of all growers’ profit in equation (15) subject to conditions (2) to (5) and constraint (14)

(15) $$W = \mathop \sum \limits_{i = 1}^N \mathop \sum \limits_{ t= 0}^\infty {\rho ^t}\left[ {{P_t}{Q_{it}}\left( {{\theta ^{ - 1}}\left( {{\delta _{it}}} \right),{S_{it}}} \right) - {C_{it}}\left( {{\delta _{it}}} \right) - w{h_t}\left( {\theta _{it}^{ - 1}\left( {{\delta _{it}}} \right),{z_{it}}} \right)} \right]\;$$

Therefore, the discrete current value Hamiltonian is as follows:

(16) $$\mathop \sum \limits_{i = 1}^N \;\left[ {{P_t}{Q_{it}}\left( . \right) - {C_{it}}\left( . \right) - w{h_t}\left( . \right) + \rho {\lambda _{i,t + 1}}\left( {{G_{it}}\left( . \right) - {S_{it}}} \right)} \right]\; - {\tau _{i,t}}\left[ {\mathop \sum \limits_{i = 1}^N {S_{it}} + \mathop \sum \limits_{i = 1}^N {h_{it}} - \overline {{D_t}} } \right]$$

where ${\tau _{i,t}}\;$ is the shadow price associated with the plastic mulch waste constraint (14), which represents the minimum monetary incentive the social planner has to impose on grower i so that the total amount of plastic mulch pollution in the farmland soil ( ${S_{it}})$ and the plastic waste sent to disposal facilities ( ${h_{it}})$ from all growers, N, will not be over the limit $\overline {{D_t}} $ . Therefore, ${\tau _{i,t}}$ can be considered as a corrective tax. By solving the first-order conditions of discrete current value Hamiltonian, if the constraint (14) is binding, we find that the optimal corrective tax is defined as

(17) $${\tau ^w} = {{{\rm{\bar P}}\Psi \left( . \right) - {C_\delta } - {\rm{w}}{h_\delta }} \over {{h_\delta }}}$$

Note that if the plastic mulch waste generated by all growers are less than the limit, $\mathop \sum \nolimits_{i = 1}^N {S_{it}} + \mathop \sum \nolimits_{i = 1}^N {h_{it}} \lt \overline {{D_t}} $ , the optimal corrective tax is ${\tau ^w} = 0$ . Since the models for the individual grower and social planner cannot be completely solved analytically, we turn to simulation methods to solve for the optimal solutions to the models. Figure 1, Box D illustrates that once the social planner determines the corrective tax amount, growers will incorporate the tax into their decision-making, internalizing the social cost of plastic pollution and considering both farmland and landfill impacts. The numerical methods and results are discussed in sections “Empirical model and data, Results and discussion.”

Illustrative case study: deterministic environment

Grounded in the optimization specification and Karush-Kuhn-Tucker equations (1)-(11), and following Schnitkey and Miranda (Reference Schnitkey and Miranda1993) and Miranda and Fackler (Reference Miranda and Fackler2004), we apply the functional forms:

(18) $${Q_{it}} = {\alpha _0} - {\alpha _1}{\delta _{it}} - {\alpha _2}{S_{it}} - {1 \over 2}{\alpha _{11}}\delta _{it}^2 - {1 \over 2}{\alpha _{22}}S_{it}^2$$

(19) $${h_{it}} = {\beta _1}{z_{it}} - {\beta _{12}}{\delta _{it}}{z_{it}}$$

(20) $${C_{it}} = {v_0} + {v_1}{\delta _{it}}$$

(21) $${G_{it}} = {\eta _0} - {\eta _1}{\delta _{it}} + {\eta _2}{S_{it}} - {\eta _3}{z_{it}} - {\eta _{13}}{\delta _{it}}{z_{it}} + {\eta _4}$$

(22) $$w = \tilde w*\varphi, \;\;where\;\;\varphi = {\chi _0} + {\chi _1}z_{it}^*\;$$

The well-defined functions allow us to numerically illustrate the results. ${\alpha _0}$ represents the crop production under the PEMs. The negative coefficients of the production function ${\alpha _1}$ and ${\alpha _{11}}$ captures the impact of the changes in degradation rates on crop production. Meanwhile, negative coefficients ${\alpha _2}$ and ${\alpha _{22}}$ measure the reduced crop production associated with the increased plastic residue in the farmland soil. The total amount of the plastic mulch used by grower i during period t is denoted as ${\beta _1}$ . ${v_0}$ and ${\eta _0}$ mean the production cost not associated with the plastic mulch and the amount of plastic mulch residue in the farmland soil at the very beginning of period t, respectively. ${\eta _3}$ captures the impact of disposal decisions on the dynamic accumulation of plastic pollutant in the farmland soil. ${\delta _{it}}{z_{it}}$ reflects the interaction between the grower’s disposal decision and the mulch’s degradation rate, which may in turn affect the plastic pollutant in the farmland soil. ${\eta _4}$ represents a base level of residue from PEM (or BDM for that matter) whether the mulch goes to a landfill or stays on the farmland. $w$ is the economic disposal cost, and it is determined by the disposal cost in monetary value, $\tilde w$ , and the effort coefficient $\varphi $ . $\varphi $ is a function of the portion of mulches to be removed from the farmland, $z_{it}^*$ . When near zero portion of mulches is to be removed, then the effort coefficient ${\chi _0} \gt 0$ . As the portion of mulch required to removed increases, the effort coefficient will increase with the rate of ${\chi _1} \gt 0$ . We use effort coefficient to capture the situation that as the grower seeks to remove more percentage of plastic mulch from the farmland to the landfill, the more effort they will put in, which increase the economic disposal cost.

We apply the collocation method to solve the current dynamic economic model (Bellomo et al Reference Bellomo, Lods, Revelli and Ridolfi2007; Golub and Ortega Reference Golub and Ortega1992). Under the deterministic environment, we perform sensitivity analysis to examine how changes in key parameters affect the decision maker’s optimal choice and the accumulation of plastic residue in the farmland soil. Results are presented in section “Results and discussion.”

Data

This study focuses on tomato growers due to readily available data, their extensive use of mulch, and the significance of tomato production within specialty crops.Footnote ¹³ Tomatoes rank among the top three vegetables in the United States by area harvested and total production, contributing over 50% of the total vegetable category in 2019 (USDA-NASS 2020). Furthermore, their economic importance is undeniable, placing them among the top three vegetables in terms of value, alongside head lettuce and onion, accounting for 32% of the total vegetable production value in 2019 (USDA-NASS 2020). Additionally, mulch plays a critical role in tomato production. To simplify analysis, we assume a one-acre tomato farm and normalize all inputs and outputs on a per-acre basis. While this study focuses on Washington State tomato growers, the particular numerical values might not be directly applicable to a specific grower, as well as other crops or regions, due to differences in crop prices, labor costs, and disposal costs. However, the modeling framework with sensitivity analyses, such as examining the impact of landfill tipping fees and the influence of crop price changes on optimal choices, offer valuable insights that can be generalized to a broader context.

In the baseline analysis, average tomato prices are $1.4/ lb, which is based on the northwestern US tomato retail price (USDA 2021).Footnote ¹⁴ The plastic waste disposal fee is assigned $0.042/lb, which is based on the surveyed landfill tipping fee in Washington in 2018 (Environmental Research and Education Foundation 2018).Footnote ¹⁵ A field survey was also conducted to characterize tomato growers’ plastic mulch choices in Washington by Mount Vernon Northwestern Washington Research and Extension Center’s Specialty Crop Research Initiative (SCRI) project team from 2010 to 2012 (Galinato et al Reference Galinato, Miles and Ponnaluru2012). All the baseline parameter values are presented in Table 1, and we assume the discount factor $\rho $ is equal to 0.9.Footnote ¹⁶ Piehl et al (Reference Piehl, Leibner, Löder, Dris, Bogner and Laforsch2018) reported about 0.1 pounds of plastic residue per acre in German agricultural land, whereas Gao et al (Reference Gao, Yan, Liu, Ding, Chen and Li2019) reported over 200 pounds of plastic residue per acre in China on lands that have had long-run exposure to plastic mulch use and its residue. Our baseline assumptions and initial conditions of plastic residue are more in line with Piehl et al (Reference Piehl, Leibner, Löder, Dris, Bogner and Laforsch2018), meaning we do not impose in the baseline that the farmland has a history of long-run buildup of plastic residue in the soil.

Table 1. Baseline coefficients of production, cost, disposal, and plastic residual evolution in a fresh-market tomato production system

¹ The following values of parameters are obtained from “2011 Cost of Producing Fresh Market Field-Grown Tomatoes in Western Washington.”

${\alpha _0}$ representing the crop production under the PEMs is 30,360lb/acre.

${\beta _1}\;$ denoting the total amount of the used plastic mulch used during period t by grower i. The parameter value is obtained from Galinato et al (Reference Galinato, Miles and Ponnaluru2012).

${v_0}$ representing the production cost not associated with the plastic mulch used by grower i is equal to $25,982.92. The production cost has been adjusted based on the agricultural price paid indexes in the USDA NASS Quickstats database. Machinery cost in 2019 is 24% higher than in 2011; and all other production items (commodities, services, interest, tax, wage rates) in 2019 is 10.5% higher than in 2011.

² Values are calibrated based on Galinato et al (Reference Galinato, Miles and Ponnaluru2012) and Jiang et al (Reference Jiang, Ma, Li and Zhang1998). The elasticity of soil plastic residue in affecting production is less than -0.028% when plastic pollutant levels are 3.2 lb. Following their work, we assume parameter values, ${\alpha _1},{\alpha _2},{\alpha _{11}},{\alpha _{22}}$ that result in an elasticity of less than 0.028%. While there is no study quantifying the relationship between mulch degradation rate and cost, we assume parameter values, ${v_1},$ and conduct sensitivity analysis to check the robustness of the results.

³ The remaining parameter values are assumed to satisfy the first-order conditions and the functional form assumptions in section “Private grower optimization.” Due to the lack of empirical studies providing values for these parameters, we conduct sensitivity analysis using different parameter values to check the robustness of the results. This paper presents sensitivity analysis for key parameters; a complete sensitivity analysis is available upon request.