Research

INTRODUCTION

Environmental policies often create negative externalities (Nordhaus 2019). One of the most prominent examples is leakage (Lima et al. 2019, Meyfroidt et al. 2020), where the benefits achieved by a policy in one region (e.g., conservation) are offset by unintended consequences in another (e.g., deforestation). Leakage is increasingly recognized as a critical challenge for assessing the effectiveness of greenhouse gas mitigation policies. This issue is particularly salient in the implementation of Reducing Emissions from Deforestation and Land Degradation (REDD+) programs (Streck 2021), where deforestation is often displaced to other locations, whether at local (Harrison and Paoli 2012), national (Fischer et al. 2016), or international scales (Atmadja and Verchot 2012). However, the occurrence and magnitude of leakage vary considerably across cases (Naughton-Treves et al. 2005, Albers and Robinson 2011, Geldmann et al. 2018, Fuller et al. 2019, Ford et al. 2020, Eisenbarth et al. 2021).

For instance, Ford et al. (2020) examined 120 tropical and subtropical forest-protected areas and found evidence of leakage in 55 cases. Among these, 78% experienced higher deforestation levels than predicted in counterfactual scenarios without protected area designation. Conversely, the remaining 65 cases showed no evidence of displaced deforestation. This near 50-50 split in the occurrence of leakage highlights the complexity of evaluating leakage, underscoring the need for a better understanding of its drivers and consequences.

Whereas part of the challenge in determining whether a policy creates leakage involves improvements in measurement and statistical procedures (Ewers and Rodrigues 2008, Fuller et al. 2019, Ford et al. 2020, de Assis Barros et al. 2022), the other part of the equation involves a better theoretical understanding of why some policies create harms beyond their jurisdictional boundaries and others do not. To date, most theoretical work has focused on general equilibrium models that aim to capture the macro-economic forces that produce leakage (Murray et al. 2004, Gan and McCarl 2007, Burniaux and Oliveira Martins 2012, Filewod and McCarney 2023). In contrast, there is a relative scarcity of microeconomic theoretical work examining the impact of leakage on the welfare of individuals and communities residing in areas where spillovers occur. Even less attention has been given to how agents on the periphery adapt to the damages created by leakage (but see Gan and McCarl 2007, De Geest et al. 2017, De Geest and Stranlund 2019, Andrews et al. 2024). A better understanding of these processes is crucial for conservation planning and for designing policies that reduce damages outside their jurisdictional boundaries, thereby increasing their overall efficacy (Ford et al. 2020).

This paper aims to formalize the micro-scale social-ecological processes involved when conservation policies induce activity-shifting leakage, where users relocate their activities beyond the policy’s jurisdiction (Albers and Robinson 2011, Henders and Ostwald 2012). Activity-shifting leakage, unlike market leakage (Filewod and McCarney 2023), occurs when resource users, or “bandits” (Olson 1965), shift their exploitation to areas outside the regulated zone (Murray et al. 2004, Lima et al. 2019). Our focus is on understanding how such leakage affects the economic and institutional behaviors of peripheral communities and, in turn, the ecological, economic, and institutional dynamics of these areas. For example, do peripheral communities develop governance institutions to manage and protect their resources, thereby deterring encroachment? Alternatively, do they shift their own resource use to more distant areas, thereby displacing costs further? Or do they adapt by entering labor markets and altering wage dynamics? Our analysis centers on systems typical of small-scale, nature-based offset programs in low- and middle-income countries, where agents often rely on de facto open-access resources, have limited connections to global markets, and operate under weak institutional frameworks.

Advancing theory in this domain requires addressing three key components. First, a causal understanding of the damages inflicted by leakage on peripheral communities is needed. Second, we must develop a framework linking these damages to adaptation strategies at individual and community levels. Finally, we need to catalog and mechanistically describe these adaptation strategies to elucidate their implications for system-level dynamics. To this end, we employ a bio-economic modeling framework, which integrates economic behavior with ecological processes (Gordon 1954, Sethi and Somanathan 1996, De Lara and Doyen 2008, Andrews et al. 2024). This approach allows us to examine the mechanisms through which leakage generates damage and to formalize the processes of adaptation.

On a practical level, our work shows that activity-shifting leakage does not always result in additional environmental or economic harm. For example, leakage may increase the number of resource users without necessarily exacerbating deforestation rates. The extent of environmental damage depends on factors such as the resource’s pre-leakage stock levels and its open-access equilibrium. Additional factors include the production and profit functions of both those causing the leakage and those residing in peripheral areas. Even in cases where environmental damage is minimal, economic damages to peripheral agents may still occur, such as through changes in labor market dynamics. Although mitigating some damages, adaptation strategies can also impose costs on more distant communities, especially in regions with weak governance structures. However, leakage may also catalyze the development of stronger property rights, underscoring its complex implications.

This study is not a review of leakage as a concept nor a descriptive account of its empirical manifestations. Instead, it focuses on the formal modeling of leakage dynamics. In the following sections, we introduce the bio-economic time allocation model as a tool to study leakage, analyze the types of damages it causes, link these damages to adaptation strategies, and explore the system-level dynamics of specific adaptive behaviors. See Table 1 for descriptions of variables and parameters.

THE BASIC BIO-ECONOMIC MODEL

Consider a two-group model where each group comprises N agents managing an associated renewable resource. At the beginning of each period, agents allocate their labor between harvesting the resource (e) and participating in wage labor (1 − e). Under baseline conditions, agents are restricted to their group’s boundaries, and each group operates independently. The objective for each agent is to maximize income (π), which derives from two distinct sources: resource harvesting and wage labor. This is represented as follows:

(1)

The first term on the right-hand side denotes income from resource harvesting, determined by the price of the resource (p) and the harvested quantity (h). The harvested quantity is influenced by labor allocated to harvesting (e) and the resource stock level (B). The second term represents income from wage labor, with 1 − e indicating time spent in non-harvesting economic activities and w representing the exogenously determined wage rate.

Harvesting function

Assuming a Cobb-Douglas production function, the harvesting function can be expressed as:

(2)

Here, q represents total factor productivity (e.g., “catchability”), while α and β denote the elasticities of labor and resource stock, respectively. Aggregate harvesting within a group, H, is the sum of individual harvests (h) for all N agents:

(3)

Resource dynamics

The dynamics of the resource are governed by the balance between natural regeneration and harvesting. Changes in the stock level depend on the remaining stock after harvests (B(H)), the intrinsic growth rate (r), and the carrying capacity (K):

(4)

Open-access equilibrium

With a mechanistic way to link harvest to resources, we can now recover a well-known dynamic of these systems, which is critical for understanding how leakage affects communities. In the absence of governance institutions and under the assumption of unit elasticities (α = β = 1), the open-access equilibrium stock level (B_oae) can be calculated by observing that, given a sufficiently large labor supply, the system will equilibrate when economic rents from the resource are zero. This equilibrium can be derived as:

(5)

This equilibrium, often referred to as the “tragedy of the commons,” reflects the stock level where the resource stabilizes under unregulated harvesting. Notably, B_oae depends solely on economic factors (w, p, q) rather than biological parameters (r, K).

System dynamics

Figure 1 illustrates the fundamental dynamics of this system. In panel A, the y-axis represents revenue from harvesting (pH), while the x-axis denotes aggregate labor effort dedicated to harvesting (E):

(6)

In Figure 1A, the red lines represent the opportunity cost of labor, determined by the wage rate (w). The intersection of these lines with the profit curve (pH) determines the B_oae, the point where marginal opportunity costs equal marginal gains from harvesting. This equilibrium produces the equilibrium effort E_oae and the equilibrium harvest value pH_oae. However, the maximum economic yield occurs at the effort level where the vertical distance between the profit curve and the cost line is maximized, which typically lies to the left of pH_oae.

In Panel B, the relationship between aggregate effort and stock level is shown under unit elasticities for two separate cost curves. As opportunity costs decrease, the equilibrium stock value (B_oae) also declines, reflecting the reduced cost of harvesting and its impact on resource stock levels.

The system dynamics implicitly assume that any given group has a sufficiently large population to supply the theoretical amount of labor needed to reach the open-access equilibrium (B_oae). However, in many real-world systems, particularly in small-scale societies like those examined in this paper, this assumption may not hold. These communities may lack the labor capacity to push the system to B_oae, resulting in an equilibrium with a higher stock level and greater aggregate payoffs. This situation persists until additional users enter the system and exert further pressure.

Moreover, when production potential is constrained by factors such as catchability (q), labor elasticities (α, β), or when opportunity costs are high (w), the stock level may remain relatively undisturbed. Consequently, the apparent sustainability of many open-access systems (Moritz et al. 2018) may stem more from low population densities, rudimentary technology, or limited market access than from any deliberate or intrinsic management practices.

LEAKAGE AND DAMAGES

When a policy is introduced in one group, activity-shifting leakage (Schwarze et al. 2002) occurs if agents travel to another (peripheral) group to harvest. In essence, leakage is simply additional harvesting, with the key distinction being the identity of those doing it. Thus, we define these harvests as H_l as:

(7)

Here, E_l represents the aggregate effort applied to harvesting by bandits, while B remains the stock value of the resource in the peripheral group. In this context, leakage can be understood as an increase in the aggregate labor supply within the system.

However, when these bandits travel to the peripheral group to harvest, they can cause damages from the perspective of the peripheral agents. These damages are defined as the loss in utility experienced by peripheral agents when comparing a counterfactual scenario with no leakage to one where leakage exists at a non-zero level.

To model total damages as a function of leakage, we consider three distinct types of damages: environmental, direct, and indirect. Let D(H_l) denote the total damages as a function of the amount of leakage into the system. We define each component type of damage separately, where environmental damages (which concern a loss of utility caused by the loss of the resources) are represented by D_E(H_l), direct damages (which directly harm people’s earnings through lower harvests because of a loss of resource stock) by D_D(H_l), and indirect damages (which harm people’s earnings through losses to income from other sectors) by D_I(H_l).

(8)

Finally, for notational simplicity, assume that {D_E, D_D, D_I} ∈ A, as this notation enables us to express total damages as a composite of different damage types, each of which may vary uniquely with the level of leakage H_l.

Environmental damages

Environmental damages (D_E) are directly concerned with the loss of a resource, independent of its impact on earnings. These damages can be expressed as:

(9)

Here, U is a utility function. The first term represents the resource stock in the absence of leakage (H_l = 0), while the second term represents the stock under conditions of non-zero leakage (H_l > 0).

Crucially, environmental damages are not connected to the economic valuation of the resource stock. Instead, they depend on the resource stock (and perhaps quality) alone; therefore, they encompass nature’s intrinsic, non-utilitarian values, such as biodiversity, cultural significance, and ecosystem services. These values are often omitted from traditional economic models (including the bio-economic model above) but remain central to conservation efforts. For instance, forests may hold existence value for global stakeholders irrespective of their extractive or recreational use, thus allowing for a broader understanding of the impacts of leakage beyond direct resource exploitation. Although these damages are not explicitly part of the bio-economic model presented above, they can be conceptualized as losses in stock, irrespective of their impact on earnings.

Direct damages

Direct damages (D_D) pertain to the economic losses borne by individuals or communities as a result of reduced resource stock levels. These damages specifically affect agents’ earnings on the periphery, whose payoffs decrease as leakage reduces as the resource stock B(H_l). Direct damages can be formulated as:

(10)

In this expression, π(h(B(H_l|H_l = 0))) denotes the payoffs from harvesting under no leakage, while π(h(B(H_l|H_l > 0))) represents the payoffs under leakage. Note that because effort allocated to harvesting is a function of the stock level, leakage also indirectly implies a change in labor allocations across sectors, (e = e(B(H_l))), - generally with more labor being allocated to other sectors when stocks are depleted because of leakage.

Relationship between environmental and direct damages

Although environmental and direct damages are related through stock levels, B(H_l), the presence of one does not necessarily imply the presence of the other. The extent of these damages depends directly on the interaction between bandits and peripheral agents. Specifically, environmental damage is heavily influenced by the relationship between the pre-leakage stock level (B) and the open-access equilibrium (B_oae), as well as the economic and production conditions faced by the two groups.

To illustrate, consider three scenarios. In the first scenario, the peripheral group has an open-access resource and a low population, N, which, even at maximum exploitation, does not have the production potential to drive the resource to the open-access equilibrium (B > B_oae). This situation could represent horticulturalists in the Amazon basin, where their ability to exploit the forest is constrained by their productivity. Imagine a conservation program implemented in a neighboring village with the same production technologies, market prices, and labor efficiency. The conservation program introduces some non-negligible amount of activity-shifting leakage, leading to additional harvests by bandits in the peripheral group’s territory. In this case, there can be substantial environmental damages D_E, which additionally cause direct damages to income from lost harvests. If H_l is large enough, the extra harvests can drive the resource stock to the open-access equilibrium. We can see this process in Figure 1A by a rightward movement along the x-axis caused by the bandits’ additional labor added to the labor pool.

In the second scenario, the peripheral group starts with a larger population, and the resource is already at the open-access equilibrium (B = B_oae). Activity-shifting leakage from the conservation program introduces additional users. However, surprisingly, the resource stock remains unaffected as long as both groups share identical production functions and market conditions. Why does this happen? Although more users are exploiting the resource, the stock does not decline because individual harvesters, including the bandits, adjust their efforts to maximize their payoffs. This behavior prevents the aggregate labor input from exceeding the break-even point where marginal costs equal marginal gains.

The key to this coordinated behavior lies in the shared equilibrium. When the resource stock dips below B_oae, the opportunity costs of harvesting outweigh the benefits, causing all users to reduce effort. As the resource regenerates slightly above B_oae, users resume harvesting cautiously. Over time, this dynamic ensures the resource stock stabilizes at B_oae. In this scenario, while direct damages occur—due to payoffs being split among more users (the peripheral agents plus the bandits)—environmental damage is negligible because the stock does not fall below the original B_oae.

Now, consider a third scenario where the profit functions of bandits differ from those of peripheral users. A conservation program might alter market conditions, resulting in changes to prices, wages, or production efficiencies (Filewod and McCarney 2023). If bandits have lower wage rates, higher prices, or more efficient production technologies, the “open-access equilibrium under leakage” (B_oael) will equilibrate at a lower stock level. The shift in equilibrium can be expressed as:

(11)

Here, Δq, Δα, Δp, Δw, ΔN represent differences in productivity, elasticities, prices, wages, and population size (aggregate users) between leakage and non-leakage conditions. Figure 1 illustrates the impact of differing wage rates on equilibrium values. The lower cost line shifts the equilibrium from B_oae1 to B_oae2, while also reducing revenue. By introducing heterogeneity into the system, bandits continue harvesting beyond the break-even point for peripheral users, driving the equilibrium to a lower stock level (B_oae2).

Indirect damages

At this point, we anticipate the reader’s concern, questioning the omission of other damages, such as indirect effects encompassing harm to ecosystem services or downward pressure on wages caused by the mass entry of bandits into the wage-labor market. These types of damages, which indirectly affect payoffs through changes in income w(H_l) from sectors other than harvesting, are collectively referred to as indirect damages (D_I).

(12)

Given the diverse nature of these costs, we model two distinct types of indirect damages caused by leakage to illustrate how these processes work.

Labor market dynamics

First, let us consider the indirect damages arising from labor market dynamics. The wage rate (w) is determined by the interaction of supply (the number of workers seeking employment) and demand (the number of available jobs). Ceteris paribus, an increase in the labor supply in non-harvesting sectors caused by banditry, shifts the supply curve outward without a corresponding increase in the demand curve, resulting in a lower wage rate.

This dynamic can be approximated by assuming that the wage rate is derived from the marginal productivity of labor, expressed as:

(13)

Here, γ represents the elasticity of labor on productivity, and ω is a scalar that determines the absolute wage rate. The magnitude of γ influences how wages respond to changes in the labor supply in the wage labor market; lower values of γ result in greater downward shifts in wages as the labor supply increases.

Figure 2 shows the consequences of these indirect damages on the system’s dynamics. In Figure 2A, the solid black line represents revenue from harvests, as before, while the new dotted, lighter black lines depict total profits at different levels of γ. The topmost dashed line corresponds to conditions where wages remain unresponsive to changes in labor market entry, while the lower line reflects increasing downward pressure on wages as more people enter the market.

As above, we assume that an increase in aggregate labor (E - a rightward movement along the x-axis) is primarily driven by a rise in banditry. As banditry increases, a larger share of labor from the peripheral agents is reallocated to other sectors, exerting downward pressure on wages. From Panel A, we observe that when γ < 1, wages decrease in response to market entry, indicating a marginal reduction in harvesting costs. Consequently, as banditry increases, less labor is allocated to other sectors due to the depressed wages, leading to a higher allocation of labor toward harvesting. In Panel B we see that this ultimately results in environmental degradation, as the B_oae shifts to the right because of the lower marginal costs.

Ecosystem service dynamics

Next, let us consider damages to earnings caused by a loss in ecosystem services. Here, the extent of these damages is tied to the resource stock (B), with causal pathways often mediated by biophysical changes. For example, deforestation can lead to increased sediment runoff, which may suffocate coral reefs, reduce fish populations, and subsequently decrease earnings from fishing. To model this, the wage function can be expressed as a decreasing function of the resource stock (B), carrying capacity (K), and a sensitivity parameter (ρ):

(14)

Although not shown here, the system’s dynamics are nearly identical to the market-entry case described earlier. However, the causal pathway through which leakage affects wages differs. In this scenario, as leakage increases and leads to stock losses (B), the resulting loss of ecosystem services exerts downward pressure on wages (w), causing additional indirect damages through reduced income from other sectors. At the same time, lower wages increase the relative value of harvesting, creating a negative feedback loop that results in a lower B_oae and reduced profits.

ADAPTATIONS AND DAMAGES

Given the diverse types of damages caused by leakage, we now build a framework that integrates micro-scale adaptation strategies with macro-level damages and adaptation costs to find optimal responses to the damages imposed. This theory defines the marginal costs and benefits of adaptation strategies, identifies optimal switch points, and calculates both cumulative and marginal impacts. By doing so, it allows for the determination of optimal investments in adaptation strategies and the allocation of resources to minimize adaptation costs and leakage-related damages, thereby maximizing social benefits (see similar approaches in Nordhaus 2019, Ueckerdt et al. 2019, Glanemann et al. 2020).

Micro-scale adaptations

Individuals and communities can allocate resources x to each strategy, with x_s indicating the investment in a specific strategy. Each strategy s ∈ S has a marginal benefit function MB_s(x_s), representing leakage abated per unit investment, and a marginal cost function MC_s(x_s), representing the cost of achieving an additional unit of investment.

To measure damages abated by each strategy (D_As), we calculate the difference between the integral of its marginal benefit from the baseline leakage level (H_l0). This integral represents the leakage abated, which is then used to calculate and sum the various types of damages across all categories:

(15)

Simplifying, we calculate the marginal damages abated:

(16)

Finally, subtracting the marginal costs (MC_s) from the marginal damages abated (MDA_s) yields the marginal net benefit of a strategy:

(17)

Given the marginal net benefits of all strategies, we can determine the optimal allocation of resources x across these strategies. To do so, we track each strategy’s marginal net benefit as investments increase. Starting with the strategy offering the highest MNB, agents allocate resources until its MNB equals that of the next-best strategy, such that:

(18)

Where s indicates the current best strategy and s + 1 indicates the next best strategy. When the marginal net benefit of two strategies meets, we have a switch point. This ensures that no additional benefit can be gained by reallocating resources. Let {x₀^*, x₁^*, ...x_n^*} be the set of optimal switch points, where x₀^* = 0 and x_n^* = x, the total amount of resources invested.

Figure 3 illustrates the dynamics of these adaptation strategies. Panel A shows the marginal net benefits for three hypothetical strategies (s₁, s₂, and s₃). The optimal switch points (x₁^* and x₂^*) indicate where investment (x) should be reallocated to other strategies to maximize net benefits. Panel B shows the total leakage abated as a function of optimal investments.

Given that we know the optimal switch points, the total leakage abated (y) by all strategies at a fixed level of investment is a piecewise sum of integrals up until the fixed point x and is computed as:

(19)

Where x_j and x_j + 1 are the bounds for the j-th segment and s_j is the strategy selected for the j-th segment.

The cost of adaptation (C(x)) for a given level of investment is obtained by integrating the marginal cost function, MC_sj(x), across all strategies in the same fashion as above:

(20)

Macro-scale adaptations

At a macro scale, the optimization challenge involves identifying the optimal investment level, x^*, to minimize the total costs of adaptation (C) and leakage damages (D). Figure 4A illustrates these costs and damages (refer to Equation 18 and 19) as functions of the remaining leakage in the system. The remaining leakage, L, is calculated as the difference between the baseline level of damages when no adaptation investment is made (x = 0) and the total damages abated at a positive investment level (L = H_l(x ∣ x = 0) − H_l(x ∣ x > 0)).

In Figure 4A, the solid line represents damages, which decrease as leakage is reduced. The dashed line represents adaptation costs, which increase as investments in adaptation are made. Together, these curves depict the trade-off between reducing damages and the costs of adaptation.

The optimal investment level (x^*) minimizes total social costs (SC), which is the sum of adaptation costs (C(x)) and damages (D(x)):

(21)

This optimal value, x^*, is the level of investment at which the combined costs adaptation and damages are minimized, ensuring the most efficient allocation of resources and can be seen in Figure 4B. Moving back to Figure 4A, at the optimal invest level x^*, the remaining leakage is H_l^*, and the total costs are comprised of C^* (adaptation cost) and D^* (remaining damages).

Figure 4C shows that this optimal point corresponds to the intersection where the marginal cost of adaptation equals the marginal cost of damages:

(22)

Meaning that further investment would no longer provide a net benefit.

Beyond x^*, additional investments result in diminishing returns, as the incremental cost of further adaptation exceeds the corresponding reduction in damages. Conversely, insufficient investment leads to unrealized benefits, where leakage damages remain unnecessarily high because of a lack of sufficient adaptation measures.

Once the optimal investment level (x^*) is determined, it becomes straightforward to revisit the set of adaptation strategies (S) (see section Micro-Scale Adaptations) and apply the outlined procedure to calculate the optimal mix of strategies. By doing so, agents or communities can, in theory, ensure that resources are allocated effectively across different options, maximizing the overall benefit of the adaptation efforts while managing costs and residual leakage.

One of the fundamental features of these abatement curves is that they highlight the trade-offs involved in adapting to leakage. Although a group could theoretically attempt to eliminate all leakage, the associated adaptation costs could be prohibitively high. Thus, agents settle for abating some damages and absorbing the remainder as a net loss.

ADAPTATION STRATEGIES

The section above outlines how to calculate optimal investments in adaptation given a set of strategies with known marginal costs and benefits. However, identifying the full set of adaptive strategies, let alone their costs and benefits, is non-trivial because of the strategies’ dependence on pre-existing ecological, cultural, economic, and institutional arrangements. As a result, cataloging such adaptation strategies requires, at least in its early stages, a case-study-based approach to assess their extent and variability.

Transitioning from general theory to system-specific details presents inherent challenges. This shift reduces generalizability and entails an epistemological gamble, requiring the researcher to assert that the specified functional forms accurately represent genuine causal mechanisms. However, such specificity becomes feasible when there is a sufficient understanding of the system, whether obtained through ethnographic, deductive, or statistical methods.

Leveraging our research in Pemba, Tanzania, we investigate a limited set of adaptation strategies commonly observed in a system where households derive up to 30% of their income from forest goods (Andrews and Borgerhoff Mulder 2022). In this context, programs aimed at incentivizing forest conservation (Andrews and Borgerhoff Mulder 2024, Andrews et al. 2021) are generating leakage across local administrative boundaries (Borgerhoff Mulder et al. 2021; Clark et al. 2024a). This situation has spurred the adoption of forest management institutions designed to deter banditry (Andrews et al. 2024).

Here, we formally examine a limited set of adaptive strategies employed by small-scale communities and individuals to mitigate the damages caused by leakage. To do so, we assume that the aggregate harvest is negligible relative to the total market supply and that the policy-creating leakage does not affect price levels. By making these assumptions, we exogenously specify price levels and exclude considerations of market leakage.

Boundaries

Elinor Ostrom’s first principle for collective governance (Ostrom 1990, Andrews et al. 2024) serves as an illustrative adaptation to leakage: communities can establish and enforce clear social boundaries to regulate resource access. The enforcement of these boundaries becomes essential when a community is subject to leakage from a neighboring group.

Well-maintained boundaries provide significant benefits, such as preventing leakage, thereby safeguarding resources and facilitating the implementation of sustainable management policies without external interference. However, controlling access incurs costs: funds must be raised for patrols, equipment, and guards. Boundary maintenance represents a public good, subject to the free-rider problem, where benefits accrue to all users, but the costs are borne privately.

In well-institutionalized settings, governments or community-based user groups can determine the desired level of leakage reduction, estimate the costs of wages, equipment, and infrastructure, and allocate budgets or raise taxes accordingly (Muneepeerakul and Anderies 2020). Such institutions mitigate the cooperation problem by having infrastructure that routinely provisions public goods.

In contrast, most small-scale communities affected by leakage lack strong centralized institutions. Individual community members are often responsible for establishing and maintaining boundaries. To encourage contributions to this public good, specific sub-strategies or incentive structures can be deployed that enhance collective adaptation (Ostrom 1990, Wunder 2008, Blom et al. 2010, Atmadja and Verchot 2012, Streck 2021, Andrews et al. 2024).

One such strategy is to use seized goods or collected fines as a form of “wage,” creating individual-level incentives for investment in boundary maintenance. However, fines and seizures differ in their effectiveness at reducing damages. Seizures require the actual harvesting of goods, potentially causing environmental damage. In contrast, trespassing fines can prevent leakage damages before resources are exploited.

The aggregate payoffs in this context can be expressed as:

(23)

In the Appendix, we calculate how changes in investment affect earnings from seizures p(∂z∂x_s) as:

(24)

The function comprises two main components: the direct and recursive contributions. The direct component of the function represents the immediate, straightforward effect of investments on seized goods. This term, ∂z∂x_s, reflects how boundary investments translate directly to a higher probability of capturing bandits and seized goods without accounting for any indirect or feedback-related pathways.

The recursive contribution accounts for the indirect, interconnected pathways through which changes in investment affect the earnings from seized goods. Specifically, it models how investments alter leakage (H_l) and resource stocks (B), which in turn feed back into the profitability of seized goods. The term ∂z ∂H_l shows how profits from seized goods increase with leakage as seizures increase. However, this effect is mediated by the relationship ∂H_l ∂z, which captures how seizures themselves reduce leakage by discouraging banditry, creating a negative feedback loop similar to Lotka- Volterra-like dynamics.

The recursive contribution also incorporates the dependence of leakage on the resource stock through ∂H_l ∂B. Larger resource stocks can attract more leakage due to their higher potential profitability for bandits. Yet, investments in boundaries (x) indirectly increase resource stock because banditry decreases, as indicated by the term ∂B}∂x, further modulating the overall impact on seized goods.

The denominator of the recursive term, ∂H_l ∂z ∙ ∂z∂H_l, represents a feedback loop between leakage and seized goods. This ensures that the recursive term dampens the likelihood of any runaway effects. Additionally, given that the sign of the second term is negative, the recursive causation can create cycles under specific parameter values, with leakage periodically oscillating between high and low levels because of the dynamics mentioned above.

Secondary leakage

Suppose that borders are either entirely unviable because of their institutional cost or are insufficient to deter the tide of leakage. In such a scenario, what alternatives are available to people? One solution is to adopt the role of a “bandit” oneself, traveling to neighboring groups to harvest and generating a cascade of secondary leakage. Secondary leakage does not stop environmental or direct damage. Instead, secondary leakage aims to directly recoup some of the lost profits caused by leakage, thereby abating damages by creating downstream externalities for others. Consider a situation where the focal population has an additional neighbor (who is not the source of the leakage) and also possesses a similar open-access natural resource (represented by subscript ₂). Agents in the peripheral group can choose to allocate a proportion x_s of their harvesting effort to travel to the neighboring group for harvesting, incurring a cost c(x_s). As such, their payoff function becomes:

(25)

When agents adjust x_s, they directly influence the distribution of effort between the two resource locations. As x_s increases, more effort shifts to the neighboring resource, which raises the aggregate harvesting effort in neighboring group e(x_s) while decreasing the focal group’s aggregate effort e(1 − x_s). This shift in allocation affects both the neighboring harvest, h₂(e(x_s), B₂(x_s)), and the focal harvest, h(B(x_s),  e(1 − x_s), H_l). Specifically, increased effort e(x_s) in the neighboring group initially leads to higher harvest levels, while diminishing returns eventually reduce the marginal benefit from additional effort in this group. Conversely, as effort e(1 − x_s) in the focal group declines, the focal harvest decreases, but at a diminishing rate, which can mitigate the total loss of focal harvest associated with reallocating effort to the neighboring group. This interdependence between the two harvests forms a core trade-off in the system.

Additionally, changes in x_s influence the resource stocks B(x_s) and B₂(x_s) in both groups. As effort directed toward the focal resource declines with increasing x_s, the focal resource stock B(x_s) tends to increase because of reduced harvesting pressure. However, the focal resource is also influenced by leakage H_l, which increases as the remaining stock level increases B(x_s). This constrains the growth rate of B(x_s), influencing the focal group’s immediate and long-term harvest potential. For the neighboring resource stock B₂(x_s), higher effort e(x_s) may initially reduce the stock because of harvesting. However, as diminishing returns from additional effort reduce the harvest benefits in the neighboring group, the marginal incentive for allocating further effort may decline, potentially stabilizing or even allowing recovery of B₂(x_s) if agents begin to allocate less effort toward the neighboring resource.

Given that the cost function c(x_s) is assumed to be increasing at a positive rate with increased investment, the travel costs will limit the extent to which agents can reallocate effort before costs outweigh benefits. This means an optimal allocation of x_s exists where the marginal gains from reallocating effort are maximized relative to the rising travel costs.

Although secondary leakage can help alleviate some of the costs imposed by primary leakage, it does so by generating additional externalities beyond its own group. This can potentially trigger a cascade of leakage, where each successive group contributes to creating damages for their more distant neighbors. In this context, the strategy’s viability hinges on the balance between travel costs and the benefits derived from harvesting in distant neighboring communities. It is important to note that secondary leakage is not a Pareto-superior or even a neutral outcome. It is a situation where the self-interest of individuals can lead to net losses in the system as they initiate a cascade of leakage.

DISCUSSION

This paper introduces a theoretical framework to study the relationships between leakage’s damages and the adaptation strategies used to deal with these. In doing so, we have illustrated some critical system dynamics caused by this adaptive process that can help inform the empirical literature: (1) The magnitude of environmental and direct damages in peripheral areas is contingent upon the relationship between the pre-leakage stock level of the resource and its open-access equilibrium. If the pre-leakage stock level is above the open-access equilibrium because of production constraints, leakage can cause environmental and direct damages. (2) If the pre-leakage stock level is already at the open-access equilibria, then the extent of direct damages depends on the bandits’ marginal costs and benefits. If bandits’ production and profit functions are the same as agents in the periphery, no environmental damage is expected but other damages may be caused; however, when bandits have a higher valuation of the resource or are more efficient harvesters—then, we expect loss to the resource stock and all subsequent damages. (3) There are various forms of indirect damages, which are context-specific, which can affect the well-being of agents in the periphery via circuitous routes not directly linked to the resource stock level. (4) A theoretical equilibrium point (well known in climate change economics) delineates the optimal resource allocation for abating leakage damages based on the costs and benefits of available adaptation strategies. (5) Finally, the adaptive strategies employed by agents on the periphery can generate wide-ranging economic and ecological impacts. These include cyclical changes in deforestation rates and institutional quality, and cascading economic and environmental damages that extend to ever more distal groups.

Our research provides some practical recommendations for researchers who study leakage. Firstly, if there are no measurable changes in deforestation (environmental damages), it does not necessarily indicate that there are no other damages produced by activity-shifting leakage that affect the welfare of individuals and communities in the periphery. Therefore, when evaluating conservation project outcomes, focusing only on environmental damages caused by leakage might be overly narrow. The extent of leakage detected by many studies could underestimate both the proportion of cases with leakage and the extent of harm incurred by leakage, as leakage is often only determined from satellite imagery, thereby capturing only environmental damage.

Bio-economic models such as ours implicitly suggest that one of the primary adaptation strategies is labor reallocation to alternative sectors, also known as livelihood diversification. In our case, diversification is merely making the best of a bad situation. Given the existence of indirect damages operating through wage-labor dynamics, it is essential to investigate changes in income (both subsistence and labor), labor force participation, and wage rates to determine whether leakage produces other damages to economic well-being. However, diversifying livelihoods often comes at other costs, not only in expected net income but also in time and resources spent on retraining. These costs associated with retraining and labor reallocation are heterogeneously expressed along well-explored pathways of age, gender, and education that conditionally affect people’s ability to participate in alternative livelihoods and their expected wages. Therefore, given the pervasiveness of such potential damages, researchers should pay more attention to the social and economic damages caused by leakage, particularly in sustainability contexts, where the traditional emphasis has been placed solely on bio-physical outcomes.

Many of the differences in outcomes observed in the leakage literature could be explained by a better understanding of the inter-relationship between stock levels and economic incentives. It is particularly important to recognize that the existing ecological status of the resource, combined with its economic value to all user groups, plays a critical role in determining the extent and types of damage caused by leakage. Fundamental to this process is understanding the system’s initial state prior to leakage—whether it is already at its open-access equilibrium or has not yet reached it because of labor constraints. However, part of the challenge here is that estimating the open-access equilibria requires much work. The open access equilibrium is a theoretical quantity that largely depends on parameters such as prices, wages, and production technology, which are rarely stable. A second major challenge lies in estimating such elasticities in often only partially monetarized economies. Only with such data can we bring modern inferential tools to estimate open-access equilibria via methods such as approximate Bayesian computation (Clark et al. 2024b). With the development of methodological improvements to estimate such latent quantities, we can improve our ability to predict whether a conservation policy will likely produce leakage by focusing on how agents behave with the set of options available to them.

Given the importance of these “outside options,” additional qualitative and case study research explicitly concentrating on what is happening in the peripheral areas where activity-shifting leakage is displaced is essential. Satellite images alone will not provide a complete picture of how leakage is being played out in the real world, particularly when groups of people already live in these peripheral areas and thus produce dynamic, socially determined outcomes. Here, rich ethnographic descriptions of the individual and political responses and system-wide dynamics are critical to the general development of theory and inference in the study of externalities.

In an attempt to help this process along, we have initiated systematic, albeit limited, documentation of various adaptation strategies employed by individuals and communities residing on the periphery of policy areas to mitigate the negative impacts of activity-shifting leakage. This documentation is crucial for understanding the wide variation in whether or not a conservation policy produces leakage.

For example, establishing boundaries that penalize outsiders from harvesting their resources may enable peripheral communities to deter activity-shifting leakage. Alternatively, given the exact structure of the payoff functions, leakage could create cycles of deforestation that only occur when investment in border patrols temporarily fades. The impact of such cycles on statistical analysis is that the observed leakage at any given time may be a temporary aberration, and aggregation could obscure significant dynamics. A practical recommendation is to utilize time series data and statistical methods capable of estimating ordinary differential equations, thereby capturing these underlying dynamics (Clark et al. 2022).

Moreover, if collective action proves challenging and institutions cannot be established, primary leakage may lead to secondary leakage, displacing damages to more distant communities. This possibility underscores the importance of carefully selecting the size of buffer zones for measuring leakage. Given this, larger buffers may be preferable when agents in peripheral areas have limited alternative economic or strategic options or travel costs are particularly low.

With the possibility of secondary leakage, one obvious extension of our framework is considering such strategies spatially explicitly. Introducing a spatial component beyond a simple two-group setup will yield different landscape-level effects, identifying appropriate routes of travel via which leakage may occur. Other research has already begun using more spatially explicit modeling frameworks that can be easily integrated with the tools developed here (Delacote et al. 2016).

Our modeling approach differs from the typical general equilibrium models in the leakage literature (Murray et al. 2004, Burniaux and Oliveira Martins 2012, Filewod and McCarney 2023), as these models aim to estimate the amount of leakage produced by a policy at a macro-economic scale. Our goal has been to provide a mechanistic micro-economic description of adaptation to activity-shifting dynamics that can account for the welfare implications of leakage. In doing so, we have had to simplify certain aspects of our analysis, particularly price dynamics. It is well known in the economic literature that conservation policies can create leakage not necessarily through the displacement of direct harvesting but by adjusting the price of the goods and inducing supply constraints (Filewod and McCarney 2023). Although our examples focus on contexts where changes in supply are likely too small to impact prices, incorporating such dynamics may become necessary when applying these models to other scenarios.

Finally, whether or not a policy creates environmental damage is not the whole story when it comes to leakage. The adaptive strategies people use to mitigate the damages of leakage can create powerful dynamics, ultimately affecting not only the success and failure of interventions but the social-ecological system at large. Our objective has been to introduce a generic theoretical modeling framework enabling researchers to explore how the adaptive behavior of individuals on the fringes of conservation programs and environmental policies can produce far-reaching system-wide dynamics. By better understanding these processes, researchers can formulate more accurate predictions regarding the potential impacts of conservation policies and diminish the incidence of leakage. We hope this will contribute to the sustainable management of natural resources and the well-being of communities situated in the often marginalized periphery of conservation areas.

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