3.1 Manufacturing firms

The home firms $h$ comply with the climate policies and have a profit function $\Pi ^{h}$ as presented in Equation (1).

(1) \begin{equation} \Pi ^{h}=pq_h-\frac {1}{2}q^{2}_h\underbrace {-\frac {\theta }{2}z^{2}-a\,(q_h-z-c)}_{\textrm {CCM}}\underbrace {-\frac {\eta }{2}c^{2}-b\, (c)}_{\textrm {VCM}} \end{equation}

Gross revenues are the product of prices $p$ and quantities $q_h$ . The costs, which have a quadratic structure, originate from production, abatement, and offset. They include the amount of permits and offsets purchased, $z$ and $c$ , respectively, as well as their CCM and VCM prices, $a$ and $b$ . We adopt a quadratic cost function to achieve linear marginal costs in production, abatement, and offset use. Considering offsets to be nonlinear could be attributed to search costs, information asymmetries, and intermediary fees (Drechsler, Reference Drechsler2022). Intermediaries are excluded; therefore, industries incur the costs of Measuring, Reporting, and Verifying (MRV) offsets to comply with quality standards. The quadratic form synthesizes this dynamic well, being previously used in addressing competition among groups of firms, even in the case of offsets (Jones and Mendelson, Reference Jones and Mendelson2011; Wang and Wang, Reference Wang and Wang2015). $\theta , \eta$ are the efficiency of the abatement and offset processes. Fixed costs are null in home firms.

The profit function of relocating firms $r$ presents a form such as in Equation (2).

(2) \begin{equation} \Pi ^{r}=pq_r-\frac {1}{2}q^{2}_r - R \end{equation}

Companies relocating avoid climate policy costs, while facing a fixed cost $R$ . This feature could be intended as an implicit sunk cost of investing in a foreign market, as well as the bureaucratic burden of dealing with border authorities. Except for the fixed cost structure, the relocating firms present identical marginal costs to the home industry. However, these companies do not need to interact with the regulators to buy permits or with land managers for offsets.

3.2 Offset and agricultural firms

We characterize the land sector participating in offset as comprising small, homogenous companies with no ability to unilaterally influence agricultural market equilibrium. As in European agriculture, where farmers typically own less than five acres and operate family businesses (Cook, Reference Cook2020). These farmers or land managers operate under carbon-neutral premises, or under a regulation that exempts their activities in some fashion from the same monitoring rules as those of industrial firms. They can make their activities carbon-negative to produce an offset, or at least reduce their emissions from the benchmark if we consider looser regulatory standards. For simplicity, we will ignore the distinction between removal and reduction. The agricultural market considered in our model is assumed to be independent of the dynamics of companies involved in climate policies. Consequently, we have incorporated this independence implicitly into the model’s evolutionary dynamics.

The management of land for offset purposes necessitates the introduction of new practices to integrate them into mainstream activities. As the literature review highlights, these can take two main forms: substitute offsets (CO) or additional offsets (AOs), translating into different degrees of substitutability with standard practices. The profit function of the offset agricultural firms is the following Equation (3).

(3) \begin{equation} \Pi ^{o}= p_a q_o - \frac {c_a}{2}q^2_o + b s-\frac {c_o}{2}s^2 -\psi s q_o \end{equation}

With $q_o$ being the quantity of agricultural goods produced by offset firms, and $s$ the quantity of offsets. $\psi$ represents the switching cost between the two activities and is partially influenced by the literature on divisionalized firms (Hirshleifer, Reference Hirshleifer1957). When the value of the switching cost parameter is zero, the marginal cost of offsets is unrelated to agricultural products and vice versa. On the contrary, whenever it is positive, the marginal cost of offsets increases proportionally to the quantity of agricultural products. The increased cost is proportional to $\psi$ , meaning that the greater the parameter is, the more relevant the substitution cost, hence the stricter the substitutability.

If the market price of the offset is null, land managers in offsets can still use agricultural products and maintain climate-relevant production at zero, thereby avoiding switching costs. The simplification we considered here is the price homogeneity between traditional agriculture and the commercial element, offset by the value of byproducts. The offset agricultural firms of both types compete with the traditional business model. The companies within the latter approach do not present any share of land for absorbing GHG and, therefore, rely solely on the revenues of standard agricultural activities, driven by the price $p_a$ , as shown in Equation (4).

(4) \begin{equation} \Pi ^{a}=p_a q_a - \frac {c_a}{2}q^2_a \end{equation}

The profit structure from traditional agricultural activities remains unchanged in the evolutionary process. In this sense, they serve as a benchmark against which the offset activity is compared.

3.3 Market-clearing conditions

The model presents four markets, whose clearing conditions are defined simultaneously. The distributions of agents within each subgroup determine the prices by presenting their optimal supply to the market. Industrial companies also have an optimal demand in the offset market. The supply of CCM is determined exogenously by the regulator.

Commodities market: The first market presented here is where the population of $N$ industrial firms sell their commodities. As a first attempt at modeling the interaction of VCO and CL, we focus on a scenario of perfect competition, where price-taking companies of a homogeneous commodity sell optimal quantities of products. The market-clearing condition is presented here as an inverse demand function:

(5) \begin{equation} p=\hat {p}-(1-x)Nq^{*}_h-xNq^{*}_r. \end{equation}

We consider an elastic demand, where $\hat {p}$ is the reserve price. The market price $p$ follows after the firms set the optimal quantities $q^{*}_h, q^{*}_r$ , representing production in either the home or relocation market, with $x$ being the latter’s share.

Emission permits market: The production of industrial commodities generates emissions. Companies operating within the jurisdiction are required to participate in compliance markets and limit their emissions according to a limit known as the cap. This sets the second market-clearing condition:

(6) \begin{equation} (1-x)N(q^{*}_h-z^{*}-c^{*})=Q \end{equation}

The equilibrium in the allowance market is found where the fixed supply of permits $Q$ meets the demand from home industrial firms, the proportion $(1-x)$ . According to its climate objectives, the administration sets the cap on allowances at the start of the regulatory process. The demand is then formulated according to an optimal production decision. This quantity equals the emissions minus abatement $z^{*}$ and the offset quantity $c^{*}$ . VCM emerge then as buffers and, in theory, should reduce the risk of price spikes.

Offsets market: The third market-clearing condition is where offsets supply meets demand. The former corresponds to the land devoted to GHG absorption. The latter refers to the number of offsets purchased by the domestic firms subject to CCM:

(7) \begin{equation} (1-x)N(c^{*})=yN_a s^* \end{equation}

Thus, the left-hand side of this equation is the demand for offsets, and the right-hand side is the supply, which is equal to the product of the number of offsetters $yN_a$ and the optimal production of offsets $s^*$ . The equilibrium is therefore endogenous to the structure of both the agricultural and industrial sectors.

Agricultural market: The agricultural market similarly allocates commodities to the industrial market. An inverse demand function ties the agricultural prices to a reserve price $\hat {p}$ reduced by the quantities produced optimally.

(8) \begin{equation} p_a=\hat {p}_a-(1-y)Nq^{*}_a-yNq^{*}_o. \end{equation}

The demand for agricultural products is elastic in this case. Mainstream and offsetters are price-takers in this market as well. The three equations are solved in a system at each iteration and are then plugged into the firms’ profit structure. The shares $x$ and $y$ at time $t+1$ are generated using an evolutionary algorithm.

3.4 Evolutionary dynamics

The VCM and CCM markets interact through two similar replicating dynamics algorithms, à la random matching game proposed by Weibull (Reference Weibull1997). These dynamic functions calculate the portion of companies relocating at every iteration based on where more profit is generated. For example, the costs of climate policies drive the share of firms involved in carbon markets. If there are greater profits in relocating, CL will increase. The VCM diffusion is endogenous to the equilibrium between the demand side of home industrial firms and the supply side of land managers involved in offset activities. Companies transition from a traditional agricultural business model to a carbon-negative production approach only if there is a financial benefit to doing so. The markets are in equilibrium if there is indifference in the selection of the business model and the location, thus stability is found in the population shares $x$ and $y$ . The agents select their production strategies based on efficiency, but they acquire this information “slowly.”

The number of $N$ industrial firms decides whether to stay home or relocate their activities by comparing the profit differences and considering the opposing side. If staying within the jurisdiction of the climate law, the population of companies $(1-x)N$ can buy allowances or offsets. In contrast, if profits from relocation are greater, a proportion of firms will relocate. The differences in profits determine the proportion of firms relocating. The replicating algorithm is therefore equal to:

(9) \begin{equation} \dot {x}_t = x_t(1-x_t)\Delta _t \Pi ^f(x_t,y_t) \end{equation}

The profit differential of industrial firms $f$ is $\Delta _t \Pi ^{f}= \Pi ^{r}-\Pi ^{h}$ , and when it is equal to zero, the share reaches equilibrium. The group of relocating firms indirectly influences the VCM. Participation in such trade is voluntary for both agricultural and industrial firms. Allowances are sold through auctions regulated by an external authority, influencing directly the former replicating algorithm, and are fixed. The proportion of agricultural firms participating in offset production, denoted by $y$ , evolves according to the profit differentials following replicator dynamics. This value determines the supply of offsets.

(10) \begin{equation} \dot {y}_t = y_t (1-y_t) \Delta _t \Pi ^l(x_t,y_t) \end{equation}

The share of agricultural firms involved in offset stabilizes as $\Delta _t \Pi ^{l}=\Pi ^o-\Pi ^a=0$ . In synthesis, the agents will use the information available in the price signals at time $t$ to identify the production strategies which affect the profit dynamics, which in turn determine the evolutionary shares $x$ and $y$ for time $t+1$ . The markets adjust from the economic performances of $q^*_h$ , $z^*$ and $c^*$ at time $t$ and produce new signals for $t+1$ . They will adjust the supply and demand after receiving information about changes in the market structure.

Thus, the VCM is the meeting of two evolutionary processes. One resides on the demand side, with industrial firms not relocating and interested in offsetting their activities. The supply is characterized by the agricultural firms willing to participate in VCM. The equilibrium of offset practices in the agricultural sector is found when the two evolutionary processes reach stability.

Some insights into the equilibrium in carbon markets can be gained by solving each profit differential separately. The industrial firms will decide whether to move according to the cost of participating in CCM net to VCM, as shown in Equation (11).

(11) \begin{equation} \Delta _t \Pi ^f= \frac {1}{2}a^2+a(p-a)-\frac {1}{2}\frac {a^2}{\theta }-\frac {1}{2}\frac {(a-b)^2}{\eta }-R \end{equation}

The costs of abatement $\frac {1}{2}\frac {a^2}{\theta }=z^*$ and offset $\frac {1}{2}\frac {(a-b)^2}{\eta }=c^*$ must be greater than the cost of demanding CCM permits $a(p-a)=aq^*_h$ and a wedge in operation’s profitability for firms to remain in the jurisdiction of climate regulation. The relocating cost $R$ generates a burden that further deters firms from abandoning climate mitigation efforts. This profitability wedge is $\frac {1}{2}a^2-R$ . Optimal production in home firms includes carbon price, thus, barrier $R$ reduces the profitability wedge, while CCM price $a$ augments it according to the cost structure. The achievement of this condition is based on the value of the cost of relocating. In case $R\gt \frac {1}{2}a^2$ , all industrial firms will eventually participate in CCM and VCM. The alternative channel for increasing the competitiveness of local firms is a reduction in the offset price, since $\frac {\partial \Pi ^f}{\partial b}=c^*=\frac {a-b}{\eta }\gt 0$ . Such a reduction would diminish the alternative channel of emission reduction and also mitigate the demand on CCM, depressing permit prices. Policy mechanisms intended to boost the supply of offsets could have this effect. A similar reasoning can be established for the diffusion of offsetters in the agricultural sector, driven by the differences in profit from Equation (12).

(12) \begin{equation} \Delta _t \Pi ^l =-\frac {1}{2}\frac {(b-\psi p_a)^2}{(\psi -1)(1+\psi )} \end{equation}

If the differentials in agricultural profits are negative, it means that the share of carbon-negative land managers capable of supplying offsets will decrease over time. For simplicity, we will assume that the factors of marginal costs were similar across agricultural practices ( $c_a=c_o$ ), meaning that mainstream farmers and offsets activities have similar marginal costs. Since offsets must be produced by firms that have already achieved net neutrality, it is fair to assume that offset activities have similar marginal cost factors associated with carbon-neutral farm products. In this case, if the switching cost is not sufficiently low ( $\psi \lt 1$ ), there is no marginal profit wedge that can explain the diffusion of offsets without some exogenous policy driver. Conversely, the greater the distance between the marginal revenues of offsets and agricultural prices (adjusted for switching practices), the greater is the share of land managers involved in VCM. Therefore, we should expect substitute practices to be less competitive than additional ones in a system that involves perfect information and linear marginal costs of offsets and monitoring without policy intervention.

3.5 model’s solution and scenarios under different policies

The literature has presented several common elements in addressing CL. Policy mixes, for instance, tend to enhance abatement effectiveness and bolster the competitiveness of local businesses. However, the role of offsets as auxiliary policies has not yet been thoroughly examined. Notably, the relationship between offset activities and CL remains unclear, in contrast to the extensively investigated effects of free allowances. The CBAM represents another proposed policy aimed at mitigating CL. Consequently, equilibrium results for a model incorporating both CL and VCO should consider these varied policy settings.

We will then consider the role of VCO in three different scenarios: under no constraint, under a carbon tariff, and with the possibility of rationing the demand for offsets. This is because the involvement of VCM in the compliance market presents potential risk factors for the institutions and other stakeholders. We investigate the possibility of the administration limiting the use of offsets and adopting a tariff to deter firms from relocating. It is worth noting that these different settings impact the structure and closure of the baseline model presented in the previous subsections. In particular, the optimal quantities vary according to the policy in place. Therefore, in this subsection, we elucidate these differences.

3.5.3 Exogenous rationing

The last policy we implemented is standardizing offset activities for the firm. The Chinese ETS is an example of such experimentation, where 5% of surrendered permits can be generated using VCM (La Hoz Theuer et al. Reference La Hoz Theuer, Hall, Eden, Krause, Haug, De Clara, Hoz Theuer and Clara2023). For simplicity, we presumed that the amount of offsets should have been proportional to the cap of the nation and adjusted to the number of companies that have not relocated, such that:

(13) \begin{equation} \hat {s}=\frac {Qf}{N_a} \end{equation}

The coefficient $0\lt f\lt 1$ indicates the amount of emissions that offsetters will compensate and is decided by the regulator. In the end, the emissions within the climate jurisdiction will always equal $Q(1+f)$ , where $Q$ represents the regulated amount and $Qf$ represents the offsetters. Under this rule, the quantity of agricultural goods produced changes to $q_o^*=\frac {p_a-\hat {s}\psi }{c_a}$ , being the supply exogenous to the firm behavior.

4.1 Calibration

The calibration parameters for our baseline explorative solution are presented in Table 2.

Table 2.

The set of parameters for model calibration

Table 2. Long description

A table titled ‘The set of parameters for model calibration’ with six columns and eleven rows. The columns are labeled ‘Parameter’, ‘Description’, ‘Value’, and ‘Source’. The table lists various parameters used for model calibration, their descriptions, corresponding values, and sources. Row 1: Parameter, Did, 9; Description, Number of firms, N; Value, 50; Source, Antoci et al. (2019). Row 2: Parameter, T; Description, Periods; Value, 100; Source, Authors selection (see Section 4.1). Row 3: Parameter, N a; Description, Number of agricultural firms; Value, 30; Source, Authors selection (see Section 4.1). Row 4: Parameter, Q; Description, Amount of permits (CAP); Value, 50; Source, Authors selection (see Section 4.1). Row 5: Parameter, f; Description, Exogenous offset for each firm; Value, 0.2; Source, Authors selection (see Section 4.1). Row 6: Parameter, c b; Description, Variable Cost Factor for Offset; Value, 1; Source, Authors selection (see Section 4.1). Row 7: Parameter, c a; Description, Variable Cost Factor for Main product; Value, 1; Source, Authors selection (see Section 4.1). Row 8: Parameter, ψ; Description, Switching cost; Value, 3 or 0; Source, Authors selection (sensitivity analysis in Section 4.4). Row 9: Parameter, p with hat; Description, Reserve price for industrial goods; Value, 150; Source, Antoci et al. (2019). Row 10: Parameter, p a with hat; Description, Reserve price for agricultural goods; Value, 25; Source, Authors selection (see Section 4.1). Row 11: Parameter, η; Description, Efficiency in abatement; Value, 1; Source, Authors selection (see Section 4.1). Row 12: Parameter, n; Description, Efficiency in offset; Value, 1; Source, Authors selection (see Section 4.1). Row 13: Parameter, x 0; Description, Initial value for relocating firms share; Value, 0.01; Source, Authors selection (see Section 4.1). Row 14: Parameter, y 0; Description, Initial value for offset agricultural firms share; Value, 0.2; Source, Authors selection (see Section 4.1). Row 15: Parameter, R; Description, Fixed cost for relocating activities; Value, 5; Source, Authors selection (sensitivity analysis in Section 4.4).

To justify the parameter selection, we calibrated the evolutionary models consistently with Antoci et al. (Reference Antoci, Borghesi, Iannucci and Sodini2021, Reference Antoci, Borghesi, Iannucci and Russu2019) where possible. We set the remaining parameters based on three specific assumptions listed below.

1. 1. $c_o=c_a$

2. 2. $0\lt \frac {\hat {p}_a}{N_a} \lt 1$ & $\frac {\hat {p}}{N} \gt 1$

3. 3. No fixed costs of production

The first was indicated in Section 3.4, meaning that the generation of offsets costs as much as any other agricultural product in a carbon-neutral (or carbon-exempt) agricultural sector. Secondly, we presumed that the agricultural production is well developed compared to the offset activities. The second point relates to the dimensions of consumers’ reserve prices. Developed countries tend to dedicate relatively smaller shares of their income to agricultural products, while a greater proportion is allocated to commodities such as electronics, clothing, and vehicles. For this reason, we considered the relative market dimension for agricultural commodities to be less than unity. Conversely, industrial commodities had a relative market dimension greater than unity. Finally, in the third point, we omitted fixed production costs in all production systems for simplicity, unless imposed by an external policy.

For some parameters where values might particularly impact the dynamics, such as relocation or switching costs, $R$ and $\psi$ , we set a first tentative value and subsequently perform a sensitivity analysis in Section 4.4. The initial values for population shares $x$ and $y$ reflect realistic starting conditions with a majority of domestic and agricultural firms, while maintaining small nonzero fractions to enable replicator dynamics when economic conditions favor strategic switching.

Regarding the timeframe, we set $T$ such that each simulation tick should not be interpreted as a year, but rather as an abstract iteration period necessary to verify the stability of the results. Considering realistic timeframes for CCMs, we could potentially link each tick to a unit shorter than a year, such as a trimester. However, differences may exist in how different markets settle.Footnote ⁵ The results are presented by type of offset practice considered. Furthermore, the role of policies is addressed specifically in each case.

Figure 1.

Free Entry–Exit scenario phase diagram for the evolution of the share of relocating firms ( $x$ ) and offset firms ( $y$ ). Sample trajectories in red, equilibria as red dots. Panel (a): Additional offsets ( $\psi =0$ ). Panel (b): Substitute offsets ( $\psi \gt 0$ ).

4.2 Results

We denote an equilibrium as $[x,y]$ . The free entry–exit scenario is illustrated by the phase diagrams in Figure 1(a) and (b), with sample trajectories highlighted in red. We begin with the substitute case (Figure 1(b)), which exhibits simpler dynamics. Here, we observe multiple equilibrium points: $[0,1]$ is a source (indicating instability and repulsion of trajectories), $[0,0]$ is a saddle point, $[0.77,1]$ is an unstable node reachable only when $y=1$ , and $[0.66,0]$ is a stable node. The absence of costs for avoiding climate policies drives companies out of the CCM jurisdiction, leading farmers to exit the VCM as demand for offsets diminishes. The system settles at an equilibrium with high levels of CL ( $x \approx 0.66$ ) and no offset supply ( $y = 0$ ). The additional case (Figure 1(a)) presents more complex dynamics, featuring a string of stable points along the $x=0.66$ line. The main insight is that additionality allows offset firms to persist even in the absence of policy and CL. However, this outcome is highly dependent on initial population conditions. For realistic initial values of $y$ ( $y \lt 0.1$ ), the stable equilibrium point would be $[0.66,0]$ , mirroring the substitute case.

The second case (relocation fixed cost) reflects the presence of asymmetries in regulatory systems expressed as a positive relocation fixed cost of $R$ . The level of substitutability affects the number of equilibria available and the location. Figure 2(a) presents the case of $\psi =0$ , where $[0,0]$ is a source. The stable node is $[0,1]$ , meaning that all that farmers could do is simply “add” carbon removal activities. The demand for offsets is maximum, as the relocation cost deters CL and keeps companies within the jurisdiction. The equilibriums are split into four ones if substitutability is added. The stable node becomes a source, while $[0,0]$ and $[0.1,1]$ are saddle points. In this case, the stable node is $[0.064,0]$ . Substitutability makes offset less competitive than ordinary permits, making the VCM collapse even in the case of a very low level of CL.

Figure 2.

Relocation fixed costs scenario phase diagram for the evolution of the share of relocating firms ( $x$ ) and offset firms ( $y$ ). Sample trajectories in red, equilibria as red dots. Panel (a): Additional offsets ( $\psi =0$ ). Panel (b): Substitute offsets ( $\psi \gt 0$ ).

Finally, the case of rationing is discussed. If switching costs are null, such that offsets are additional, there are only two equilibrium points. Figure 3(a) indicates that $[0,0]$ is an unstable saddle point, for which the initial point would necessarily be $y=0$ . However, the stable node is $[0,1]$ as for the previous case. This implies that rationing does not reduce the number of offsetters if carbon removal activities are additional to the ordinary ones.

Figure 3.

Exogenous rationing of offsets scenario phase diagram for the evolution of the share of relocating firms ( $x$ ) and offset firms ( $y$ ). Sample trajectories in red, equilibria as red dots. Panel (a): Additional offsets ( $\psi =0$ ). Panel (b): Substitute offsets ( $\psi \gt 0$ ).

The rationing guarantees a stable level of farmers in VCM, even with switching costs. Furthermore, the CBAM deterred companies from relocating. The case of substitutability, as portrayed in Figure 3(b), presents one saddle point in $[0,1]$ and one stable point in $[0,0.17]$ . Substitutability makes offsets less competitive than ordinary permits, causing the VCM to collapse even with very low levels of CL.

4.3 Welfare analysis

It is interesting at this point to evaluate the desirability of each scenario besides just the shares of $x$ and $y$ . The welfare analysis in this paper follows the standard approach of aggregating the consumer surplus, considering economic and environmental costs. Since there are two main consumer commodities, there are two surpluses to consider in the partial equilibrium analysis. One is the industrial sector, while the other relates to the agricultural sector. Thus, the total welfare is calculated as the sum of the two welfare functions as in $\mathcal{W}=W_i+W_a$ . In both subcases, the social planner does not optimize the welfare function; rather, the outcomes emerge from the evolutionary dynamics and can be compared post hoc. The surplus in industrial commodities accounts for the economic cost of production within the jurisdiction of the climate policy. Relocated firms outsource their production, shifting costs to other locations while continuing to sell commodities in the domestic economy. Environmental costs are equated to pollution, which we simplify by assuming an emission intensity of unity, calculated as production minus abatement and offset. Consequently, abatement or emission compensation activities directly enhance welfare. Products from relocated activities are cheaper but incur the welfare cost of unabated emissions.

Specifically, the term $(q_h^*-z^*-c^*)$ in Equation (14) captures residual emissions per home firm: each unit of production generates one unit of environmental damage unless neutralized by abatement $z$ or offsetting $c$ . Relocated firms, by contrast, generate damages equal to their full output $q_r^*$ , since they operate outside the climate jurisdiction and neither abate nor purchase offsets; the factor $p-1$ in the second term reflects the net surplus from their cheaper production, discounted by the unit environmental cost.

(14) \begin{equation} W_i = N(1-\hat {x}) \left [p q_h^* - \underbrace { \left (\frac {1}{2}(q_h^*)^{2} + \frac {1}{2}z^* + \frac {1}{2}c^* \right )}_{Economic Costs} - \underbrace {(q_h^* - z^* - c^*)}_{Damages} \right ]+N\hat {x}\left [(p -1)q^*_r \right ] \end{equation}

The welfare function aggregates the net consumer surplus from agricultural products and considers the products relevant for the consumers. However, the cost of producing offset is also in agricultural products. Thus, we added the economic costs of offset activities in the welfare analysis.

(15) \begin{equation} W_a = N_a \hat {y} \left [ p_a q^*_o+bs - \left ( \frac {c_a}{2}(q^*_o)^2 +\frac {c_o}{2}(s^*)^2 +\psi s^* q^*_o \right ) \right ] + N_a(1-\hat {y}) \left [p_a q^*_a - \frac {c_a}{2}(q^*_a)^2 \right ] \end{equation}

Since we assumed that the sector was producing offsets that were already net-neutral to the climate, we assumed that the agricultural system does not generate environmental damages. We will evaluate the welfare of the stable equilibrium points where the initial conditions are irrelevant. The results are summarized in Table 3, where the various policies are represented, and the stability of the points is discussed.

Table 3.

Stability and welfare analysis (in welfare units) of equilibrium points for substitute $\psi \gt 0$ and additional $\psi =0$ offset practices in VCM

Table 3. Long description

A table with five rows and six columns comparing stability and welfare analysis of equilibrium points for different offset practices in VCM. The columns are labeled Policy, Equilibrium (x, y), W_i, W_o, and W. The rows are labeled with different policies: Free entry-exit, Relocation cost, and Offset rationing. Each policy has two rows of data. The values in the table are as follows: Row 1: Free entry-exit, 0.66,0, 211.938, 9.755, 221.693. Row 2: Free entry-exit, 0.66,0, 211.938, 9.755, 221.693. Row 3: Relocation cost, 0,1, 412.909, 21.108, 434.017. Row 4: Relocation cost, 0.048,0, 459.506, 9.755, 469.261. Row 5: Offset rationing, 0,1, 457.331, 23.166, 480.497. Row 6: Offset rationing, 0.037,0.216, 459.480, 15.480, 474.960.

Of the three policy settings, the worst-case scenario for welfare would be one of free entry and exit. Since no farmer has adopted the carbon-negative business model, the land has only one use and generates the least welfare. Outsourcing production due to CL reduces costs but increases pollution outside the jurisdiction. Fewer firms have to comply with the cap within the jurisdiction, reducing the demand for allowances and offsets, which drives the signal for abatement down to near zero.

Figure 4.

Sensitivity analysis for the relocation fixed costs $R$ parameter: (a) relocation fixed costs, additional offsets, (b) relocation fixed costs, substitute offsets, (c) exogenous rationing, additional, and (d) exogenous rationing, substitute.

The total welfare depends on the switching cost if a relocation cost is imposed. Agricultural welfare is higher when farmers adopt additional techniques in the agricultural business and populate the VCM. The switching cost decreases the utility of participating in emission reduction activity for the farmers and incentivises industrial firms to abate using mainly CCM. Interestingly, the welfare is higher in this second case, being the third highest scenario.

The imposition of supply offsets largely increases the utility of participation in VCM for farmers, guaranteeing a steady income from diversifying land use. In the case of $\psi =0$ , the diffusion of emission reduction practices encompasses the entire population of farmers, resulting in the highest agricultural welfare. In the case of substitute practices, a small but stable share of farmers produces the demanded offset. Since there is an abundance of instruments to reduce or abate emissions and little incentive to relocate, industrial welfare is the highest.

4.4 Sensitivity analysis

It is valuable to assess how changes in relevant parameters affect the equilibrium results. In this analysis, we vary the values of $R$ (relocation fixed costs) and $\psi$ (offset activity switching cost). It is important to note that this should not be interpreted as a bifurcation diagram, as we only show one stable equilibrium per experiment for simplicity. Nevertheless, this approach still allows us to draw insights beyond our base case calibration.

In Figure 4(a), where offsets are additional, we observe that both populations can coexist up to a threshold value of $R$ , beyond which the relocating population becomes extinct. This phenomenon also occurs in the substitute case, although the required $R$ value is higher (Figure 4(b)). This implies that even with relatively low relocation costs, a positive CL can still result, even in the presence of offsetters. Figure 4(c) and (d) present the same analysis for the case of exogenous rationing of offsets, focusing on additional and substitute technologies, respectively. In the additional technology scenario, when $R$ is smaller than the calibrated value, we observe two distinct sets of equilibria. For very low values ( $R \lt 1.8$ ), a positive $x$ and a zero $y$ are observed. For $1.8 \lt R \lt 5$ , $y$ returns to a full share, albeit with a positive and declining $x$ . From the calibrated point value onwards, the final states match those of the calibration. This indicates that even with additional technologies, lower $R$ costs could still lead to the extinction of offsetters. A similar situation is observed for substitute technology, where, notably, for some values of $R \gt 6$ , it was not possible to find a stable equilibrium even using numerical methods.

Next, we vary the $\psi$ parameter for each scenario, keeping $R=0$ in the free entry–exit case and at the calibrated value in the other two. In Figure 5, we observe stable equilibriums very similar to the calibration case, except for very low values of substitutability. Even low switching costs might still result in offsetters disappearing. When relocation costs are present, we also get a calibration-like scenario for most values of $\psi$ . Finally, for the exogenous rationing case, we observe a nonlinear behavior in the evolution of the final value for the $y$ population. In this case, it appears that increasing values of switching costs can support a higher share of offset activities.

Figure 5.

Sensitivity analysis for the offsets technology switching cost $\psi$ parameter: (a) free entry–exit, (b) relocation fixed costs, and (c) exogenous rationing

Moreover, we also introduce heterogeneity in the marginal costs of production by varying the coefficients between $h$ and $r$ firms. In particular, we reduce the coefficient for $r$ firms by half under the sensible assumption that relocating firms are more efficient or possess better technology.Footnote ⁶ We observe that this acts as a catalyst for the widening of profit differentials and can shift the equilibrium outcomes quantitatively, but not qualitatively, resulting in similar stability structures.

As additional checks, we test some of the settled parameters in various scenarios. First we consider the number of permits $Q$ , which could be considered as a climate policy stringency proxy. We test the variation of this parameter in the AOs scenario with no relocation sunk costs. We find that while the qualitative characterization of the equilibria remains unchanged, the more stringent the policy is, the higher the fraction of relocating firms (see Figure 6). We then vary the reserve price of the agricultural good, $\hat {p}_a$ , also in the free entry–exit scenarios, finding no discernible impact (see Figure 7). Finally, we consider the exogenous rationing of the offsets’ coefficient $f$ . We find that in the additional case there is no effect in rising or lowering this parameter (Figure 8) while there is an effect in the substitute case, where the final share of the $y$ population is slightly increasing with higher $f$ (Figure 9).

Figure 6.

Phase diagrams for the sensitivity analysis for different values of the $Q$ parameter (amount of permits): free entry–exit, additional offsets. Sample trajectories in red, and equilibria as red dots.

Figure 7.

Phase diagrams for the sensitivity analysis for different values of the $\hat {p}_a$ parameter (reserve price of the agricultural good): free entry–exit, additional offsets. Sample trajectories in red, and equilibria as red dots.

Figure 8.

Phase diagrams for the sensitivity analysis for different values of the $f$ parameter (exogenous share of offsets): relocation sunk costs, additional offsets. Sample trajectories in red, and equilibria as red dots.

Figure 9.

Phase diagrams for the sensitivity analysis for different values of the $f$ parameter (exogenous share of offsets): relocation sunk costs, substitute offsets. Sample trajectories in red, and equilibria as red dots.

4.5 An empirically anchored numerical exercise

To complement the qualitative regime analysis, we present a numerical illustration calibrated to match key empirical ratios observed in the EU carbon market. The exercise does not aim at producing quantitative forecasts; rather, it demonstrates that empirically plausible parameter configurations fall within the regime taxonomy developed in the preceding sections, and it sharpens the policy interpretation of our results. Since the model is normalized (quantities and prices are in abstract units), we anchor the calibration to dimensionless ratios that can be computed from observable data, rather than to absolute magnitudes. Table 4 presents the empirically calibrated parameters alongside their justifications.

Table 4.

Empirically anchored calibration parameters

Table 4. Long description

A table with 11 rows and 3 columns. The columns are labeled Parameter, Value, and Empirical justification. The table lists various parameters with their values and justifications. Row 1: Parameter, Value, Empirical justification. Row 2: N, 100, Rescaled from EU ETS stationary installations (EC, 2024); each model firm represents installations. Row 3: N subscript a, 60, Maintains N subscript a / N greater than or equal to 0.6, consistent with base calibration. Row 4: Q, 100, Q / N = 1 preserves the base calibration ratio. Row 5: p, 300, p / N = 3 greater than or equal to 1: relative market dimension assumption from Section 4.1. Row 6: p subscript a, 50, p subscript a / N subscript a approximately equal to 0.83 less than or equal to 1: smaller weight of agricultural goods in developed-economy expenditure. Row 7: theta subscript a, 1, Normalized abatement and offset MRV efficiency. Row 8: Policy and technology parameters (varied across scenarios), empty, empty. Row 9: psi, 0.5, Soil carbon (additional). Modest substitutability with ongoing production; yield penalties of 0-15% (Paul et al. (2023)). Row 10: psi, 2.5, Afforestation (substitute). Permanent land use change directly substitutes for crop production; psi greater than 1. Row 11: R, 8, CBAM. Amortized MRV, reporting, and certificate costs under EU CBAM (2026-2034). Row 12: R, 0, Pre-CBAM baseline. No border adjustment. Row 13: f, 0.05, Rationing. Consistent with China’s 5% VCM rule (ICAP, 2024) and possible EU offset eligibility caps.

We simulate the replicator dynamics for four empirically motivated configurations. VCM implementation without relocation costs and costly switching practices ( $\psi = 2.5$ , $R = 0$ ), for example, afforestation, VCM collapses ( $y \to 0$ ) and CL is high ( $x \approx 0.66$ ). This mirrors the documented instability of REDD + and avoided-deforestation credits. The situation does not improve if switching costs were reduced, for example, due to technological advancement. Such scenario ( $\psi = 0.5$ , $R=0$ ) determines high CL ( $x \approx 0.66$ ) with a string of potential VCM equilibria depending on initial conditions, thus depending on farmers willingness. Realistic starting shares ( $y_0 \lt 0.1$ ) lead to $y \to 0$ . The optimistic scenario for EU carbon emerges in carbon farming under the combined CBAM + VCM framework ( $\psi = 0.5$ , $R = 8$ ). CL is eliminated ( $x \to 0$ ) and VCM achieves full diffusion ( $y \to 1$ ). The problem of costly switching practices in VCM could be partially compensated with “rationing” in conjunction with CBAM ( $\psi = 2.5$ , $R = 8$ , $f = 0.05$ ). The analytical result of Appendix A predicts that $\psi \gt 1$ requires exogenous support; rationing stabilizes a small but positive share of VCM. This is the most policy-relevant scenario for EU afforestation offsets. The empirically motivated plausible scenarios are mapped to the taxonomy emerging from the equilibrium analysis in Table 5:

Table 5.

Empirically anchored scenarios: predicted equilibria and regime classification

Table 5. Long description

A table comparing scenarios of soil carbon and afforestation with and without CBAM and rationing. The table has four rows and five columns. The columns are labeled Scenario, ψ, R, f, Equilibrium (x*, y*), and Regime. The rows are labeled as follows: Row 1: Soil carbon, no CBAM, 0.5, 0, (0.66, 0), Free entry-exit, additional. Row 2: Soil carbon + CBAM, 0.5, 8, (0, 1), Relocation cost, additional. Row 3: Afforestation, no CBAM, 2.5, 0, (0.66, 0), Free entry-exit, substitute. Row 4: Afforestation + CBAM + rationing, 2.5, 8, 0.05, (0, y* > 0), Rationing, substitute.

The empirical exercise sharpens three messages that could be empirically tested in future works. The first is that soil carbon farming is viable without heavy intervention. Since $\psi \lt 1$ , soil carbon credits fall in the additional regime where VCM can be self-sustaining once CBAM deters CL. This supports the EU’s emphasis on soil carbon within the Carbon Removal Certification Framework.Footnote ⁷ The second is that afforestation offsets will require active policy support. With $\psi \gt 1$ , afforestation-based offsets are in the substitute regime and require either rationing or direct subsidy (e.g., through the Common Agricultural Policy) to sustain participation. Finally, CBAM is necessary but not sufficient. Border adjustment eliminates CL and creates demand for offsets, but the viability of the VCM supply side depends critically on the technology type. A “one-size-fits-all” offset policy risks supporting only additional practices while substitute ones, which include many nature-based solutions with high biodiversity co-benefits, will require targeted support.

4.6 Robustness check: Monopolistic competition

We extend the analysis beyond perfect competition in the industrial sector by introducing a reduced-form markup parameter $\mu \geq 0$ for industrial firms only, capturing monopolistic competition where each firm internalizes a fraction of its price impact on the residual demand curve.Footnote ⁸ When $\mu = 0$ , we recover perfect competition. The agricultural sector remains competitive, consistent with the small-farmer setting.

The markup modifies only the output FOCs ( $q_h^* = (p-a)/(1+\mu )$ and $q_r^* = p/(1+\mu )$ ), leaving all compliance and agricultural quantities structurally unchanged. Two opposing forces shape the effect on CL. On the one hand, the markup reduces per-firm output and hence the compliance burden, making home production relatively less costly. On the other hand, it also reduces the relocating firm’s profits symmetrically. The net effect on VCM participation is similarly ambiguous a priori: lower output per home firm depresses offset demand, but if fewer firms relocate, the pool of offset demanders grows.

We resolve this ambiguity numerically by sweeping $\mu$ from $0$ to $2$ across all six scenarios (results depicted in Figure 10). The central finding is that monopolistic competition acts as a smooth perturbation of the competitive benchmark: the stable equilibrium shifts continuously with $\mu$ , and the qualitative stability structure – number, type, and basin-of-attraction configuration of equilibria – is preserved in all scenarios. CL $x^*$ increases moderately with $\mu$ , as the markup erodes the compliance cost advantage of staying home. VCM participation $y^*$ is largely unaffected in the additional-offset scenarios (panels c and e of Figure 10), where it remains at $y^*=1$ for all $\mu$ tested, and declines smoothly in the substitute-offset case (panel f). Crucially, the policy rankings of the main text are preserved: relocation costs and rationing continue to outperform free entry–exit, and AOs remain more competitive than substitute ones across all degrees of market power considered.

Figure 10.

Stable equilibrium shares under monopolistic competition. Blue circles: carbon leakage share $x^*$ ; red squares: VCM participation $y^*$ . In panel (a), $y^*$ depends on initial conditions due to the non-hyperbolic manifold along $x \approx 0.66$ .