2.1. Private outcome

We let h _Lt denote legal landings at time t, while h _It denotes illegal landings. The employees determine the total harvest level, h _Lt and h _It , given the wage contract offered by the firm. With full flexibility in designing the wage scheme, the firm can make the wage scheme contingent on both legal and illegal landings, implying a wage scheme: W(h _Lt , h _It ). We assume that the employees and the firm each bear a share of the total harvest costs, which is given by c(h _Lt , h _It , x _t ). Given this, we can express the cost of the employees as αc(h _Lt , h _It , x _t ), where 0 ≤ α≤ 1 is the employee's exogenous cost share.Footnote ⁴ The harvest cost function has the following properties: $\displaystyle{{\partial c}\over{\partial h_{Lt}}} \gt 0$ , $\displaystyle{{\partial ^{2} c} \over {\partial h_{Lt}^{2}}} \gt 0$ , $\displaystyle{{\partial c} \over {\partial h_{It}}} \gt 0$ , $\displaystyle{{\partial^{2} c} \over {\partial h_{It}^{2}}} \gt 0$ , $\displaystyle{{\partial ^{2}c} \over {\partial h_{Lt} \partial h_{It}}} \gt 0$ , $\displaystyle{{\partial c} \over {\partial x_{t}}} \lt 0$ , $\displaystyle{{\partial^{2}c} \over {\partial h_{It} \partial x_{t}}} \lt 0$ and $\displaystyle{{\partial ^{2}c} \over {\partial h_{Lt} \partial x_{t}}} \lt 0$ . These properties ensure that the problem is well behaved, and they will later help us characterize the properties of the enforcement cost function. The assumptions reflect that it is costly for the employees to work on board the vessel, and higher landings and lower stock sizes require that employees exert more effort.

When a vessel exceeds the quota, it risks being penalized. Depending on the liability rule, either the firm, the employee or both may then face a fine. Let G(h _It ) ≥ 0 and F(h _It ) ≥ 0 denote employee's and the firm's fines, respectively. Both penalty functions are zero in the absence of illegal landings (F(0) = 0 and G(0) = 0), and increase exponentially with the level of illegal landings. In the base model, we assume that any value of G(h _It ) and F(h _It ) can be paid by the employee and the firm, respectively. The probability of detecting violators, γ(e _t ) > 0, is a function of enforcement effort, e _t , exerted by the regulator, but the probability is taken as given by the industry. Therefore, an employee's expected punishment if the vessel violates the quota is γG(h _It ). In the following, we analyze the behavior of a representative vessel (firm and employees). We start out by looking at the behavior of employees.

The expected payoff of a risk-neutral employee is:

Hence, the employee earns a wage, but incurs a share of the harvesting cost and must pay the expected penalty for harvesting in excess of the quota.

The firm knows the expected payoff function of the employees and can observe both legal and illegal harvests. Thus, under standard assumptions about the properties of the cost function, and a flexible wage function, the fishing firm can use the compensation scheme to induce the employees to land any given quantity (Laffont and Tirole, Reference Laffont and Tirole1993). In the base model, we impose no restrictions on the wage function and, hence, the principal can indirectly determine landings through the choice of the wage function. This makes the principal the real decision maker and therefore we turn to the optimization problem of the firm.

We assume that firms (and employees) are myopic, and do not consider the implications of their harvest decisions on the evolution of the stock. This is reasonable in a common property resource setting with many resource users, which implies that each user has only a negligible impact on the stock, and is a standard assumption in common property resource models (Clark, Reference Clark1990; Neher, Reference Neher1990). Given this, the firm maximizes profits with the parameters of the wage function as control variables. However, since the firm can indirectly determine both legal and illegal landings, we can simply use these landings directly as control variables instead of the wage function. Given that the firm faces the same probability of detection as the employees, the firm's expected penalty is $\gamma F\lpar h_{It}\rpar $ . On this basis, we can express the maximization problem of the risk-neutral firm as:

subject to

where p is the output price,Footnote ⁵ Q _t is the individual quota, and U ⁰ is the reservation payoff of the crew.

Note from the objective function (2) that the firm, just like the employees, only incurs a fraction of the total costs. Condition (3) is a quota restriction that captures that total legal landings cannot exceed the quota. This restriction will always be binding when we focus on the case where the firm harvests both legally and illegally (which is what we are interested in here). Condition (4) is a participation constraint, which expresses that agents must receive at least their reservation payoff, U ⁰, in order to participate. Thus, condition (4) implies that the firm wants the employees to work on board the vessel.

Note that since the wage function enters with a negative sign in the firm's objective function, condition (4) will always bind. This implies that:

Equation (5) shows that the wage covers the employees' reservation payoff, their costs of working on board the vessel, and their expected penalty. From this, it follows that the firm insures the agent against all costs. One might argue that the employees should be able to increase the payoff by changing the harvested quantity. However, since there are no wage frictions in the base model, the firm will design a wage contract that ensures that the employees obtain the highest possible expected payoff by doing exactly what the principal wants them to do. Consequently, changing the harvest quantities away from the firm's desired level can only reduce the employees' expected payoff. In section 3.1 we show that this is no longer the case if the wage function is constrained.

Returning to the optimization problem of the firm, we can substitute (5) into (2), yielding the following modified maximization problem:

which must satisfy the quota constraint given by (3).

Assuming a binding quota restriction, implying that the firm considers illegal landings, the Lagrangian of the maximization problem in (6) and (3) becomes:

where ε _t is a Lagrange multiplier that measures the shadow price of the quota restriction.

The first-order conditions with respect to legal and illegal landings, h _Lt and h _It , are:

Both conditions (8) and (9) state that the marginal private benefit of landings, p, must equal the marginal private costs. The marginal private cost of legal landings (equation (8)) consists of the marginal production costs of firm and employee, $\displaystyle{{\partial c} \over {\partial h_{Lt}}}$ , and the shadow price of the quota constraint, ε _t . The marginal private cost of illegal landings (equation (9)) consists of the marginal production cost, and the marginal expected penalties of firm and employee. Note that the production costs and expected penalties of both firm and employees are included because the principal fully covers the costs of the agent.

Conditions (8) and (9) allow us to derive illegal harvest as a function of the inspection rate, the quota (legal harvest) and the stock. This relationship is given by h _It  = h _It (γ, Q _t , x _t ), which is the reaction function of the firm. It is straightforward to show that, given the properties of the harvest cost function, c(h _Lt , h _It , x _t ), and the penalty functions, F(h _It ) and G(h _It ), we must have that $\displaystyle{{\partial h_{It}} \over {\partial x_{t}}} \gt 0$ , $\displaystyle{{\partial h_{It}} \over {\partial \gamma}} \lt 0$ , and $\displaystyle{{\partial h_{It}} \over {\partial Q_{t}}} \lt 0$ .Footnote ⁶

We can now derive society's enforcement cost function from the reaction function of the firm. We first invert the reaction function of the firm, which yields γ = γ(h _it , Q _t , x _t ). The inverted reaction function describes the link between the probability of being detected, illegal landings, stock size and the quota.Footnote ⁷ Next, recall that γ (e _t ) describes the relationship between the probability of being detected and enforcement effort, with $\displaystyle{{\partial \gamma} \over {\partial e_{t}}} \gt 0$ . We can invert γ (e _t ) to get e _t  = e _t (γ), and by substituting in the inverted reaction function we get e _t  = e _t (γ(h _it , Q _t , x _t )) = α(h _it , Q _t , x _t ). Finally, we introduce an enforcement cost function that depends on enforcement effort,

K(e _t ), which we assume is increasing and convex in enforcement effort: $\displaystyle{{\partial K} \over {\partial e_{t}}} \gt 0$ and $\displaystyle{{\partial^{2} K} \over {\partial e_{t}^{2}}} \gt 0$ . Now, we can substitute e _t  = α (h _it , Q _t , x _t ) into the enforcement cost function, which yields K(e _t ) = K(e _t (α (h _it , Q _t , x _t ))) = E(h _it , Q _t , x _t ). Thus, we have established that the total enforcement cost depends on the level of illegal landings, the quota, and the availability of fish. Given the properties of the firm's reaction function, $\displaystyle{{\partial \gamma} \over {\partial e_{t}}} \gt 0$ , and the properties of the enforcement cost function, we have that $\displaystyle{{\partial E} \over {\partial x_{t}}} \gt 0$ , $\displaystyle{{\partial E} \over {\partial Q_{t}}} \lt 0$ , and $\displaystyle{{\partial E} \over {\partial h_{It}}} \lt 0$ .Footnote ⁸ Thus, all else being equal, allowing for higher illegal landings implies a reduction in enforcement costs, while a larger stock leads to an increase in enforcement costs. Finally, a higher quota reduces the enforcement costs. We will make use of the properties of the enforcement cost function when we analyze the social planner case below.

2.2. Social planner problem

To find optimal aggregate illegal and legal landings at time t, h _It and h _Lt , we let a social planner maximize the present value of future resource rents for a representative vessel (firm) net of enforcement costs. Note that we include the full value of illegal landings in the welfare function, in line with Milliman (Reference Milliman1986) and Lewin and Trumbull (Reference Lewin and Trumbull1990), who conclude that the social planner should account for the value of regulatory offences in their analyses of the social value of criminal gains. Furthermore, we assume costless fines. On this basis, we can formulate the maximization problem of the social planner, with legal and illegal landings as control variables, and stock size, x _t , as the state variableFootnote ⁹ :

such that

where δ > 0 is the discount rate, and f(x _t ) is a natural growth function. In equation (10), the enforcement cost function takes into account the behavioral response of the fishing firms to changes in the quota (legal harvest) and the stock size, and the corresponding impact of these variables on the enforcement costs to achieve a given combination of legal and illegal landings. We have already established that $\displaystyle{{\partial E} \over {\partial h_{It}}} \lt 0$ . In addition, we assume that the enforcement cost function is convex in the reduction of illegal landings, $\displaystyle{{\partial^{2} E} \over {\partial h_{It}^{2}}} \gt 0$ . Furthermore, it is never optimal to eliminate all illegal harvesting: $\mathop{\lim}\limits_{h_{It} \to 0} e\lpar h_{It}\rpar = \infty$ . The resource restriction (11) states that the change in stock size $\lpar \dot{x}_{t}\rpar $ must equal the natural growth of the fish stock net of harvesting. For the natural growth function, we assume f ^′ (x _t ) > 0 for 0 <x < x _msy , and f ^′ (x _t ) < 0 for x > x _msy , where x _msy is the stock size corresponding with the maximum sustainable yield, and finally that f ^″(x _t ) < 0. A standard logistic growth function satisfies these assumptions. The problem given by (10) and (11) gives an accurate representation of how a socially optimal quota (h _Lt ) should be determined.

Efforts to hide illegal activities (avoidance) and similar costly activities related to illegal harvesting may create differences in harvest costs between legal and illegal fishing. Hence, the presence of such costs implies that we should separate legal and illegal harvests in the cost function. Furthermore, if quota enforcement did not cost anything, the social planner would eliminate all illegal harvesting in the social optimum, to avoid avoidance costs and those similarly incurred by illegal activity. However, in our model, as in real world fisheries (Arnason et al., Reference Arnason, Hannesson and Schrank2000), enforcement costs play a significant role. It typically becomes increasingly costly to raise the detection rate marginally, and it is usually suboptimal to increase the detection rate to a level that fully deters violations. We therefore focus on the case with imperfect enforcement and quota violations in the following.

The current-value Hamiltonian of the problem defined by (10) and (11) is:

where λ _t is a co-state variable that measures the marginal cost of the dynamic stock externality. Note that this marginal cost depends on the current stock size, and will thus change over time unless the fishery is in a steady-state equilibrium. Given our focus on non-compliance and liability, it is sufficient to state the optimality conditions for the control variables (legal and illegal landings):

Conditions (13) and (14) state that the marginal social benefit, p, equals the total marginal social costs of legal and illegal landings, respectively. For both legal and illegal landings, the total marginal social cost consists of the marginal harvest cost ( $\displaystyle{{\partial c} \over {\partial h_{t}}}$ , with h _t  = h _Lt , h _It ), the marginal enforcement cost ( $\displaystyle{{\partial E} \over {\partial h_{t}}}$ , with h _t  = h _Lt , h _It ), and the marginal cost of the dynamic stock externality. Note that condition (13) is the condition for the optimal quota (legal landings), while (14) can be interpreted as the condition for optimal enforcement (illegal landings). Because of a non-constant λ _t over time, the optimal enforcement policy changes over time.

Above, we mentioned that due to, for example, avoidance costs, illegal fishing might be costlier than legal fishing. However, if enforcement costs are sufficiently high for it to be socially undesirable to fully deter all quota violations, there will be an interior solution involving both illegal and legal landings. Using this, we can use equations (13) and (14) to obtain a relationship between legal and illegal landings given by $\displaystyle{{\partial c} \over{\partial h_{Lt}}} + \displaystyle{{\partial E} \over {\partial h_{Lt}}} = \displaystyle{{\partial c} \over {\partial h_{It}}} + \displaystyle{{\partial E} \over {\partial h_{It}}}$ . Hence, in the social optimum, the total marginal social cost (harvesting cost and enforcement cost) must be the same for legal and illegal landings.

2.3. Optimal liability

We can now compare the optimality conditions of the private and the social planner problems to investigate under which conditions the private solution is also socially optimal. Let us first consider legal landings, given by conditions (8) and (13). Alignment of social and private legal landings requires thatFootnote ¹⁰ :

Condition (15) implies that the firm's shadow price of the quota constraint, ε _t , must equal the marginal enforcement cost of the quota (legal landings) plus the marginal cost of the dynamic stock externality, λ _t . This will generally not hold in a decentralized fishery, but can be achieved through regulations. In the absence of enforcement costs $\left(\displaystyle{{\partial E} \over {\partial h_{Lt}}} = 0\right)$ , we obtain that ε _t  = λ _t . From section 2.1, we have that $\displaystyle{{\partial E} \over {\partial h_{Lt}}} = \displaystyle{{\partial E} \over {\partial Q_{t}}} \lt 0$ . Therefore (15) implies that ε _t  < λ _t . Thus, Q _t is larger with enforcement costs than without these costs, and this is explained by the fact that an increase in the quota reduces illegal landings and, thus, the enforcement costs.

Let us next analyze illegal landings. From the perspective of the social planner, the detection probability in the private optimality condition for illegal landings (9) depends on the exerted enforcement effort. From section 2.1, we have that enforcement effort in turn can be expressed as a function of stock size and legal and illegal landings, which implies that γ = γ(α(h _Lt , h _It , x _t )) = μ(h _Lt , h _It , x _t ). Then, by comparing conditions (9) and (14), we find that alignment of private and socially optimal illegal landings requires:

Thus, the social planner must set the expected punishment so that the marginal enforcement cost of illegal landings plus the marginal cost of the dynamic stock externality equal the expected value of the marginal penalty for both firm and employee.Footnote ¹¹ Since λ _t varies over time, we get from (16) that the optimal fine policy varies over time. We can calculate the optimal quotas and illegal landings by solving the equation system given by equations (15) and (16) for h _It and h _Lt . This, in turn, enables us to find the optimal enforcement level as a function of the optimal legal and illegal landings, and the current stock level. Finally, note that (15) and (16) represent a first-best optimum since there are no frictions in the wage function between firm and employees.

With this in place, we can investigate the implications of the allocation of liability. The mixed liability case is already reflected by equations (15) and (16). With full firm and full employee liability, respectively, either the employee pays no fine (G(h _It ) = G ^′(h _It ) = 0) or the firm pays no fine (F(h _It ) = F ^′ (h _It ) = 0). Hence, in the case of full firm liability, equation (16) reduces to:

while in the case of full employee liability, equation (16) reduces to:

Regardless of the liability allocation, equation (16) remains unchanged. Equations (17) and (18), along with equations (15) and (16), show that a social planner can achieve a first-best solution under mixed liability, full firm liability and full employee liability. Hence, the liability allocation does not matter. Regardless of whom the legal system holds liable for quota violations, the social planner can set the enforcement effort and the quota to secure an optimal outcome. The reason is the lack of frictions in the base. Hence, regardless of liability allocation, the firm can shift all relevant costs (including expected fines) fully between the firm and the employees, thereby leaving the employees equally well off regardless of the formal liability rule, at least in expectation terms.

Our main result thus far is therefore that we can reach a social optimum regardless of which actor the legal system holds liable for illegal landings. As mentioned above, the environmental economics literature reaches similar results assuming a static flow externality (Segerson and Tietenberg, Reference Segerson and Tietenberg1992; de Vries and Franckx, 2012). However, we assume a dynamic stock externality by including a dynamic resource restriction, which represents a new contribution to this literature. Indeed, our analysis shows that the dynamic stock externality, and thus the status of the resource stock, affects the optimal quota and enforcement policy. Hence, the optimal policy in our study varies over time in contrast to the static policy considered in previous work.

The lack of frictions in our model drives our liability-irrelevance result, and particularly the assumptions that firms have full flexibility in specifying wage functions. Hence, the firm can dictate the behavior of its employees, while the regulator can dictate the behavior of the fishing firms. In the next section, we consider the consequences of introducing wage frictions between firm and employee through restrictions on the wage functions, and of assuming limited financial assets for firm and/or employees that diminishes the deterrence effect of fines. We show that social optimality relies critically on the firm's ability to shift the fine penalty to the employee.

3.1. The share system of remuneration

Before we implement the share systems of remuneration in the base model, it is useful to investigate how the firm induces the employees to harvest the desired quantity. Recall that we did not do this in the analysis of the base model, since we established that with full flexibility in designing the wage contract the firm could induce its employees to harvest any quantity. Hence, we could do the analysis without analyzing exactly how the firm induced its employees to behave in a certain way.

The Lagrange function of the maximization problem of the employees is:

where u _t is the employees' shadow price of the quota constraint.Footnote ¹³ Thus, the employees also account for the quota restriction. The employees' first-order conditions for legal and illegal landings are:

Assuming, as in the base model, that the quota constraint binds (h _Lt  = Q _t ), equation (20) determines legal landings. In addition, and as discussed above, the employees' participation constraint must hold. Hence, the fishing firm can determine the appropriate compensation scheme by ensuring that the reaction functions of the employees for legal and illegal landings (equations (20) and (21)) correspond with the firm's optimal levels of legal and illegal landings (equations (8) and (9)).

With restrictions on the wage function, the firm might no longer be able to induce the employees to harvest any desired quantity. If this is the case, equations (20) and (21) will determine the harvest quantity rather than equations (8) and (9) as in the base model. Under the share-of-profit system, the employees receive a share of the firm's profits from the fishing operation, whereas they receive a share of revenues in the share-of-revenue system. We will show that while a social planner can secure a second-best optimum under profit sharing by proper choice of quota and liability allocation, only full employee liability ensures the same with revenue sharing. Assume that, depending on which share system we consider, the employees receive at least a fraction β₁ of the firm's expected profit or a fraction β₂ of the firm's revenue. In both cases, either the agent's participation constraint (4) or the wage scheme must bind. Let us assume the wage scheme binds. Then, under profit and revenue sharing, respectively, we have that:

In line with our analysis of the base model, the firm remunerates the employees based on both legal and illegal harvesting. In the case of profit sharing, this implies that the wage function includes the firm's expected penalty, γF _t (h _It ). The difference between the share-of-profit and share-of-revenue systems is that the latter does not include the firm's costs, and hence allows for no shifting of costs between the principal and the agent.

The wage functions (22) and (23) can be substituted into the firm's profit function (2). However, since the share systems leaves the firm with only one choice variable, the profit- and revenue-sharing parameters β₁ and β₂, respectively, the firm cannot dictate the harvest levels of the employees. To see this, note that the sharing parameters enter linearly into both the agent's and the principal's optimization problems, regardless of the share system. Hence, the firm can only use the share parameter to influence whether the employees work on board the vessel, but cannot use the wage incentive to influence their harvesting efforts. It follows that it will be optimal for the firm to pay its employees the lowest possible share that ensures that they will work on board the vessel, as given by the participation constraint (4), while it can only hope the employees will harvest close to the desired quantity.

With the employees determining legal and illegal landings, we must consider their decisions when deriving the conditions for optimal quotas and liability. The Lagrange function of the representative agent's optimization problem, given that the participation constraint holds, is the same as in the base model (7), but with the share wage functions (22) or (23) replacing the flexible wage function W(h _Lt , h _It ).

We can now use the employees' optimality conditions, equations (20) and (21), to determine h _Lt and h _It . Specifically, we can derive a reaction function, h _It  = h _It (Q _t , γ, x _t ), from the optimality conditions of the employees for each of the two sharing rules. Although these reaction functions will differ from the one we derived for the firm above based on the base model, it is straightforward to show that, given the properties of the harvest cost and the penalty functions, the reaction function must satisfy: $\displaystyle{{\partial h_{It}} \over{\partial x_{t}}} \gt 0$ , $\displaystyle{{\partial h_{It}} \over {\partial \gamma}} \lt 0$ , and $\displaystyle{{\partial h_{It}} \over {\partial Q_{t}}} \lt 0$ . These properties match the properties of the reaction function derived for the base model, and hence, as in section 2, we must have that $\displaystyle{{\partial E} \over {\partial x_{t}}} \gt 0$ , $\displaystyle{{\partial E} \over {\partial Q_{t}}} \lt 0$ , and $\displaystyle{{\partial E} \over {\partial h_{It}}} \lt 0$ .Footnote ¹⁴ This implies that we can compare the private outcomes to the social planner optimum given by equations (13) and (14), even though the underlying enforcement cost functions differ.

Let us now consider the optimal selection of policy instruments for the share-of profit case. Comparing the private and social optimality conditions in this case yields the following conditions:

where in (25) we have used that from the perspective of the social planner, γ = γ(α (h _Lt , h _It , x _t )) = μ(h _Lt , h _It , x _t ), as in the base model. Doing the same comparison for the share-of-revenue case yields:

Due to wage frictions between the firm and the employees, the optimal policy conditions for the two sharing rules represent second-best solutions. Let us start out by considering the optimal quota policies for the two sharing rules. Recall that in the base model, the optimal quota condition stated that the private shadow price of the quota should equal the marginal enforcement cost plus the marginal cost of the dynamic stock externality (equation (16)). However, when a sharing rule determines wages, two additional terms enter into the optimal quota condition (equations (24) and (26)): the marginal revenue and marginal costs of the firm. We must include these terms because there is no mechanism for transferring the relevant costs and benefits from firm to employees. The wage friction causes an externality where the employees ignore the effect of their harvesting choices on the firm. Thus, the social planner chooses quotas and fines to internalize this externality. There is one difference between the optimal quota policies under profit sharing (24) and revenue sharing (26). With profit sharing, the social planner only corrects for a share 1 − β₁ of the marginal harvesting cost of the firm, since profit sharing already shifts a share β₁ of the firm's harvesting costs to the employees, while we correct for the full marginal cost in the revenue-sharing case.

Let us now turn to the optimality conditions for illegal landings (enforcement), as given by equations (25) and (27). In the base model, the optimality condition for illegal landings states that the marginal enforcement cost plus the marginal cost of the dynamic stock externality should equal the marginal expected penalties of firm and employees (equation (16)). Again, with the wage frictions introduced by the sharing rules, the social planner must correct for the fact that the employees do not take into account the implications of their harvesting decision for the firm. In this case, two additional terms enter into the optimal enforcement condition, representing the firm's shares of marginal revenue and marginal costs. In addition, the marginal expected penalty that enters into the condition depends on which of the two sharing rules we consider. With profit sharing, the optimal enforcement condition (25) includes the marginal expected penalty paid by the employees directly, G ^′(h _It ), and indirectly through the wage function, β₁ F ^′(h _It ). The revenue-sharing condition (27) only includes the marginal expected penalty of the employee. As for the optimal quota conditions, we can explain these differences by the differences in underlying wage functions and, hence, for both the optimal quotas and fines, the optimal policy shifts the relevant costs from the principal to the decision-making agent. From (12) we know that λ _t is non-constant over time, and so the optimal enforcement and fine policies given by (25) and (27) must also vary over time.

Let us now investigate the implications of liability allocation between firm and employees. With revenue sharing, the optimal enforcement condition (27) contains the expected penalty function of the employee but not the firm. This implies that only full employee liability can ensure a second-best optimum. With profit sharing, the social planner can ensure a second-best optimum given by equations (24) and (25), regardless of liability allocation. While the optimality condition for quota (24) is unaffected by the allocation of liability, this is not the case for the optimality condition for enforcement (25). With mixed liability the optimal enforcement policy is given by equation (25), the term G ^′(h _It ) drops out of (25) with full firm liability, and the term F ^′(h _It ) drops out with full employee liability. Hence, the optimal quota and enforcement policy might differ depending on whether the enforcement system penalizes the firm, the employee or both for illegal landings.

Let us finally consider the special case in which the firm cannot shift expected penalties to its employees. This implies that the wage function in (22) cannot include the term γF(h _It ). This only changes the liability analysis for the share-of-profit case. Without the ability to shift a part of the expected fine to the employees under profit sharing, only employee liability ensures a second-best optimum. To see this, note that condition (24) is unchanged while βF ^′(h _It ) drops out of equation (25). Hence, firm liability cannot work. However, G ^′(h _It ) is included in equation (25), and so employee liability can secure a second-best optimum. When the firm cannot shift the expected fine to the employees, the employees do not consider βF ^′(h _It ) when making decisions about illegal landings. Hence, changes in the firm's expected fine will not affect the level of illegal landings.

In this section, we have shown that both the share-of-profit and the share-of-revenue rules can ensure a second-best optimum. However, while the allocation of liability does not matter with a share-of-profit rule, only agent liability yields the efficient outcome with a share-of-revenue rule. Hence, a sharing rule that allows the firm to share both costs and revenues with the employee makes it less critical which liability scheme the industry faces.

3.2. Limited assets

In the base model, we assume that firm and employees have unlimited assets. However, in reality both parties may face an upper limit on the fines they are able to pay, and this might change the optimal punishment and cause inefficiencies. Therefore, we now analyze the implications of asset limitations for both the firm and the employees, and discuss indemnification clauses for the employees within the modeling framework from section 2.

Let V _t and L _t denote the financial assets available to the employees and the firm, respectively, at time t. Introducing an asset constraint for the employees implies that the fine must fulfill the condition:

Note that in condition (28) the actual penalty rather than the expected penalty enters (see section 2). When interpreting equation (28), we distinguish between two cases characterized by whether the firm can offer the employee an indemnification clause. With an indemnification clause the firm covers any fines paid by the employee for illegal harvesting. Therefore, condition (28) is non-binding and, because G(h _It ) ≥ γG(h _It ), the analysis from section 2 applies. Hence, with indemnification clauses, a social planner can ensure a first-best optimum with all liability rules. Thus, provided the firm does not have a binding asset constraint, an indemnification clause is optimal in fisheries.Footnote ¹⁵ Turning to the case without indemnification clauses, full employee liability cannot secure a first-best optimum once the penalty exceeds the employee's asset constraint. This result holds even with full wage flexibility as the firm can only compensate the employees for the expected fine, not for the actual fine. However, full firm liability will work since the firm does not face a binding asset restriction. Turning to mixed liability, equations (15), (28) and (29) determine the outcome provided the employees' asset restriction in (28) is binding. In this case, increasing the employees' fine, G(h _It ), beyond the level characterized by condition (28) has no deterrent effect. Hence, once the asset constraint in equation (28) binds, a further increase in the employees' formal fine does not increase the actual fine payment and, therefore, we have that G ^′(h _It ) = 0. Next, we can find the firm's fine from condition (16) with G ^′(h _It ) = 0, which yields:

Hence, equations (28) and (29) determine the second-best optimal enforcement policy with mixed liability, while equation (15) determines the optimal quota.

Similarly, with an asset constraint for the firm, the optimal enforcement policy must satisfy the following two conditions:

Note first that if the actual penalty can be shifted from the firm to the employee, condition (30) never binds, which means that it does not matter who is held liable (see section 2). Next, if the firm cannot shift the fine to the employee, which is perhaps the more plausible case for the fishing industry, full employee liability will still work, since this case is unaffected by the asset constraint of the firm. With full firm liability, the fine should be set to the maximum allowed by the asset constraint (30), but this will generally not yield the optimal solution. However, under mixed liability, we can achieve a second-best optimum by ensuring that equations (30) and (31) hold simultaneously along with the optimal quota condition (15).

Finally, if both firm and employees face binding asset constraints, equations (28) and (30) determine the fines. In this case, the fines will be too low to ensure the social optimum regardless of indemnification clauses or wage contracts; hence, irrespective of the liability rule (firm, employee or mixed), the social planner cannot secure an optimum. Thus, in situations where the firm and employees have limited assets, we can conclude that enforcement policies are ineffective. We have seen examples of this in real-world fisheries, for example, in cases when owners of ‘flag of convenience’ vessels arrested for illegal fishing abandon their vessels rather than paying the fines (Agnew, Reference Agnew2000).