Additionality in Theoretical von Thünenian Models of Deforestation and Conservation Payments

Abstract

Simple theoretical von Thünenian models of deforestation and agricultural expansion have been extensively studied in the literature but have not yet been adapted to reflect contemporary conservation paradigms, such as the emergence of REDD+ (Reducing Emissions from Deforestation and Forest Degradation) initiatives, related payments for forest conservation, and payments for ecosystem services (PES) more broadly. We revisit Angelsen’s 1999 seminal adaptation of the 1826 von Thünenian model of deforestation and agricultural expansion and propose a “toy model” to incorporate the potential revenues from conservation payments and build on the concept of additionality in the payments for environmental services literature. As theorized, our extended model illustrates how such payments are more effective when they approach the profit margins of geographically peripherical crops that replace the forest. Moreover, it illustrates how conservation payments influence the agricultural frontier while quantifying the avoided deforestation area.

1. Introduction

In the late 1990s and early 2000s, numerous theoretical models on the economics of tropical deforestation were proposed in the literature [1,2,3,4]. Among these, Angelsen’s [4] seminal study on household deforestation has garnered considerable attention, serving as both the theoretical foundation and hypothesis justification for numerous econometric and simulation modeling studies focused on the drivers of land-use change and the effectiveness of conservation instruments and policies [5,6,7,8,9,10,11].

Classical von Thünenian assumptions have long been central to many land-use and deforestation models [12,13,14,15,16]. These assumptions are predicated on several key factors: transportation costs of agricultural products and timber progressively increasing (while land rent progressively decreases) with greater distance from the market; profit maximization behavior; homogeneous landscape (e.g., uniform in terms of soil quality and terrain); and no changes in transportation infrastructure or technological advances through time [17,18,19]. Angelsen’s seminal models of agricultural expansion and deforestation [4] adopt traditional von Thünenian assumptions [18] with additional considerations of household behavior or objectives, labor market integration, and tenure regimes [4]. Nevertheless, they do not account for potential revenues from payments for conservation or environmental services (PES), such as those offered by REDD+ (Reducing Emissions from Deforestation and Forest Degradation, plus the roles of conservation, forest management, and enhancement of carbon stocks) initiatives.

Building on the ongoing development of REDD+ and other PES programs worldwide [20], we propose an extension to Angelsen’s theoretical von Thünenian models of deforestation and profit maximization [4]. This “toy model” serves as an educational exercise, akin to Hartman’s [21] extension of the seminal Faustmann model [22] on the determination of optimal harvest age in forestry activities, thus providing a framework for exploring the theoretical constructs and practical implications of PES payments. Toy models play a critical role in scientific inquiry more broadly by offering clarity, insight, and tractability in understanding the behaviors of complex systems. It is the simplifying assumptions of these models that enable the exploration of “what-if” scenarios and the testing of theories. More specifically, in land system sciences, toy models have been used to explore the global consequences of land use [23], understand climate change [24,25], and explore complex human–environment interactions [26,27]. Toy models are also effective teaching tools because they make abstract concepts accessible to students and researchers. Our extension of Angelsen’s theoretical von Thünenian models integrates the influence of conservation payments on the determination of the optimal agricultural frontier with the concept of additionality [28]. Our extension of Angelsen’s theoretical von Thünenian models integrates the influence of conservation payments on the determination of the optimal agricultural frontier with the concept of additionality [28].

2. Angelsen’s Profit-Maximization Model Under Perfect Market Integration

Angelsen’s models [4] focus on deforestation caused by the expansion of agricultural frontiers. They are especially applicable to farm households in developing countries whose land-use decisions disregard environmental regulations. These households are of particular interest to PES and REDD+ programs because they represent a key stakeholder group [29,30] that is often associated with increasing rates of deforestation [31]. The models address distinct deforestation contexts, where decisions about agricultural expansion are modeled as investment decisions. They are also strictly focused on the quantification of deforestation associated with the expansion of cultivated areas and do not differentiate specific land-use options, such as cattle ranching and perennial crops, nor any revenue associated with the presence of forests (e.g., timber, agroforestry, non-wood forest products, etc.).

In Angelsen’s model [4] of profit-maximization behavior under perfect labor market conditions and secured land tenure regimes [4], crop yield per area unit is represented by an exogenous variable ($x$), associated with a static technological level, and the total agricultural land area is represented by$H$. For simplification, Angelsen [4] considered the production unit price to be exogenous and set at $1. Hence, agricultural production ($X$) is given by:

$$X=xH$$

(1)

Following the von Thünenian model [18], forest is assumed to be abundant and homogenous and the deforestation frontier is calculated based on distance costs from a central point, representing the household’s village (Figure 1), while agricultural land expands as a concentric circle from the village center. The area of the agricultural land ($H^T$) is thus given by:

$$H^T=\pi {\left(b^{max}\right)}^2=\underset{0}{\overset{b^{max}}{\int }}2\pi bdb$$

(2)

where, $b$ is the distance from the village centroid to the edge crop area and $b^{max}$ represents the maximum economically viable distance for cultivation (i.e., the agriculture-forest edge). Land area is assumed to be equally distributed among $N$ households, represented by$\pi {\left(b^{max}\right)}^2/N$:

$$H={\displaystyle \frac{\pi }{N}}{\left(b^{max}\right)}^2=\underset{0}{\overset{b^{max}}{\int }}hbdb;h\equiv {\displaystyle \frac{2\pi }{N}}$$

(3)

We assume that all production is sold in the market. The model includes two types of labor cost: (i) forest clearing and field preparation for cultivation (assumed to be equal to $1) and (ii) travel cost per unit of land ($q$) from the centre of the village to the forest edge ($b$). Hence, total labor is given by:

$$L=\underset{0}{\overset{b^{max}}{\int }}\left(1+qb\right)hbdb$$

(4)

Under perfect market integration, the model assumes that labor can be sold or hired at the market wage, with no subsistence requirements affecting production totals (as households can meet their dietary needs through markets). Consequently, the deforestation frontier is expanded with an increase in $x$ (agricultural price or productivity) and/or a decrease in costs ($1+q$) or wage ($w$). The model can be expressed as a profit maximization problem with respect to$b^{max}$:

$$\Pi =X-wL=x\underset{0}{\overset{b^{max}}{\int }}hbdb-w\underset{0}{\overset{b^{max}}{\int }}\left(1+qb\right)hbdb$$

(5)

where the frontier of deforestation is then given by:

$$b^{max}={\displaystyle \frac{x}{qw}}-{\displaystyle \frac{1}{q}}$$

(6)

3. An Extension Angelsen’s Model with Conservation Payments and Additionality

Extensions of theoretical microeconomic models integrating environmental services with natural resource management decisions are commonly found in the literature [32,33,34]. In this section, we propose a similar type of extension for Angelsen’s model described in Section 2, based on PES proportional to the avoided deforestation, following the REDD+ and additionality concepts [28]. In other words, conservation payments are equivalent to the difference between the optimum deforestation that can be derived from Angelsen’s original model [4] (Equation (6)), and the new optimum level in the presence of the payments (Figure 1).

Let$R$ represent the PES (or the REDD+ payment for conservation) per unit area of forest (or the forest’s carbon stock) that would otherwise be converted to crops under the optimum scenario without the PES program (with program costs assumed to be constant and incorporated into$R$). Under the profit maximization assumption, ${b^{max}}^{\prime }$ is defined as the optimum profit from the area cultivated plus the conservation payments, which is a function of $b^{max}-{b^{max}}^{\prime }$, where $0\le {b^{max}}^{\prime }\le b^{max}$. The fixed cost, assumed to be 1 in Angelsen’s original work, is replaced by the variable $f$. The new profit maximization function can then be framed as:

$$Max\Pi =x\underset{0}{\overset{{b^{max}}^{\prime }}{\int }}hbdb-w\underset{0}{\overset{{b^{max}}^{\prime }}{\int }}\left(f+qb\right)hbdb+R\underset{{b^{max}}^{\prime }}{\overset{b^{max}}{\int }}hbdb$$

(7)

with the new deforestation frontier (${b^{max}}^{\prime }$) expressed as:

$${b^{max}}^{\prime }={\displaystyle \frac{x-wf-R}{qw}}={\displaystyle \frac{x-R}{qw}}-{\displaystyle \frac{f}{q}}$$

(8)

As intended, Equation (10) demonstrates the direct impact of conservation payments on defining the optimal deforestation frontier. The deforestation frontier is expanded with an increase in agricultural price or productivity ($x$) or a decrease in costs ($f+q$), wage ($w$) or conservation payments ($R$), and vice versa.

4. Numerical Simulation

Following Angelsen [4], we conducted a sensitivity analysis to illustrate how changes in the extended model parameters affect the magnitude of the optimal deforestation frontier. Parameter values, intended solely for illustrative purposes to demonstrate the model’s mechanics, are mainly based on farm household data collected from an old deforestation frontier in the state of Rondônia, Brazilian Amazon [35] (

Table 1. Initial model parameters for the sensitivity analysis.

Parameter	Notation	Value ($)	Source
Crop income per area unit *	$x$	252.17	Caviglia-Harris [35]
Wage (adjusted per crop area)	$w$	25.35 †	Caviglia-Harris [35]
Travel cost per distance unit	$q$	0.10	Angelsen [4]
Forest clearing and cultivation cost per area unit	$f$	7.43 ‡	West et al. [6]
Conservation payment per area unit	$R$	150.00 §	Authors

* Replacing variable ‘crop yield per area unit’ ($x$) in Equations (7) and (8). ^† Given the average household off-farm income of $3887.01, six adults per household on average, and an average crop size of 51.11 hectares per household. ^‡ Given an initial value of $50.00, applying a 10% annual discount rate over 20 years. ^§ Value adopted based on the range of estimates for the region from West et al. [36].

).

We present the results with and without this extension for comparison (Figure 2). Under the given parameter settings (Table 1), the optimal agricultural frontier ($b^{max}$) was calculated as 124.62 distance units ($R=0$). Under a payment scenario of $150.00 ($R$), the new frontier decreased by almost 50% to 65.45 distance units (${b^{max}}^{\prime }$). For the most extreme payment scenario in our sensitivity analysis ($R=300$), the new frontier decreased by approximately 95% to 6.28 distance units. Still, PES ($R$), along with the forest clearing and field preparation for cultivation cost per area unit ($f$), were the least sensitive variables affecting the optimal agricultural frontier. The latter frontier was most sensitive to changes in wage ($w$), ranging from 0 to 1915.11 distance units, followed by travel cost ($q$), ranging from 32.7 to 1246.27 distance units, both displaying non-linear behaviors with decreasing returns to scale (Figure 2). As expected, increasing crop income (or yield) per area unit ($x$) was directly correlated with an expansion in the agricultural frontier.

5. Discussion

Angelsen [4] examined the causes of tropical deforestation contrasting population-driven and market-driven approaches. His work emphasizes the role of deforestation as an investment strategy, particularly in areas where clearing land secures property rights. This market-driven approach differed from prior population-driven models that viewed deforestation as resource degradation [37,38]. He showed that policy interventions supported by population-driven models (e.g., land titling and agricultural intensification) could unintentionally exacerbate deforestation in market-based systems. For example, agricultural intensification intended to reduce deforestation could make land more valuable, prompting further deforestation [39,40]—although the opposite has also been reported [41,42]. Angelsen [4] concludes that policies that promote infrastructure development (including roads) could unintentionally increase deforestation, but policies that target employment and income opportunities could reduce pressure at the forest frontier. The toy model outlined in this study shifts the focus of the original model to investment programs based on PES and the concept of additionality [28]. Following Angelsen’s original work [4], the proposed extension was characterized by easily interpretable equations under specific market conditions that can serve as an educational tool, as well as a basis for future research focused on land use, conservation, and PES [6,9,36,43,44].

Current evidence largely aligns with Angelsen’s [4] deforestation models. Economic factors including rising commodity prices [5], rural credit [45], and infrastructure development, such as roads [13,46,47], have been found to be the primary drivers of deforestation. Deforestation has also been found to be shaped by governance, policy interventions, and conservation efforts, including PES [48,49,50,51]. Evidence shows that while PES can slow deforestation rates and promote conservation [52,53,54], broader economic forces continue to drive land-use change worldwide [5,55,56].

Relatedly, the cost-effectiveness of forest-related PES as a climate change mitigation strategy has long been recognized and examined in the literature [34,57,58,59]. However, the concept of additionality has not always been present in PES studies or programs [32,60,61]. Additionality is defined as “the difference in service provision between the with-PES scenario and the without-PES baseline” [28], which is captured in our extended model as the difference between $b^{max}$ and ${b^{max}}^{\prime }$. Such a theoretical framework ensures that farm households are proportionally compensated based on the extent of avoided deforestation achieved at the specified level of conservation payment. This issue is particularly relevant for REDD+ discussions focused on commercializing carbon credits from avoided deforestation [16,20], given that the credits are not legitimate without additionality [62,63,64].

Following Angelsen [4], our proposed model serves as a simplified representation of the complex issue of deforestation. It intentionally disregards many variables to allow for easier theorization and interpretability, aiming to be “simple, but not too simple” [65]. For example, the time component is critical to deforestation studies and is often at the core of complex economic deforestation models and scenario simulations [6,33,66,67,68]. And, following the traditional von Thünenian assumptions, our model extension also ignores the biophysical and political heterogeneity of real, dynamic landscapes and their intricate relationship with agricultural profitability and deforestation pressures [51,69]. Other critical PES issues, such as leakage (i.e., deforestation displacement as opposed to reduction) are also not included in our model [39]. While more complexity arguably brings models closer to reality, they often become limited in terms of providing easily interpretable frameworks for educational purposes and stakeholder discussions.

6. Conclusions

We propose a simple theoretical extension of Angelsen’s seminal von Thünenian model of deforestation and agricultural expansion based on the profit maximization behavior of farm households under perfect labor market integration [1] to incorporate the effects of PES based on the avoided deforestation, such as those promoted by REDD+ programs. The extended model demonstrates that these payments are most effective when they approach the profit margins of geographically peripheral crops that replace forests. Furthermore, it highlights how conservation payments influence the agricultural frontier and quantifies the area of avoided deforestation in the form of a toy model, adhering to the concept of additionality emphasized in the environmental services literature [28].