Quota Enforcement in Resource Industries: Self-Reporting and Differentiated Inspections

Abstract

Quotas are frequently used in the management of renewable resources and emissions. However, in many industries there is concern about their basic effectiveness due to non-compliance. We develop an enforcement model of a quota-regulated resource and focus on a situation with significant non-compliance and exogenous constraints on fines and enforcement budget. We propose a new enforcement system based on self-reporting of excess extraction and explicit differentiation of inspection rates depending on compliance history. We use differentiated inspections to induce firms to self-report excess extraction. This system increases the effectiveness of the quota by allowing the regulator to implement a wider range of aggregate extraction targets than under traditional enforcement, while ensuring an efficient allocation of extraction. In addition, inspection costs can be reduced without reductions in welfare.

Appendix 1

1.1 Deriving Aggregate Catch Levels

We start out by presenting the parametrization of the theoretical model used in the simulations. In the following subsections, we show how aggregate catch levels are calculated under traditional and self-report based enforcement.

1.2 Model Parametrization

The parameter values used in the simulations are given in Table 3.

Table 3 Parameter values

Recall that the cost parameter $\alpha$ is uniformly distributed over the interval $[\underline{\alpha }, \bar{\alpha }]$. Based on the parameter values from Table 3, there are $n=100$ agents with cost parameters ranging from $\underline{\alpha }=75$ to $\bar{\alpha }=125$.

The interval for the intrinsic growth rate $h$ given in Table 3, represents the range of growth rates we analyze. Note that there is no variation in the growth rate, instead we evaluate the performance of the self-report based system over a range of intrinsic growth rates representing different types of fisheries. At low growth rates, there is a high need for enforcement to meet the aggregate catch objective. As we increase the growth rate, the need for enforcement declines until $h=\bar{h}=1.0217$, when no enforcement is needed. In this case, the aggregate catch target is reached under open access.

1.3 Traditional Enforcement System

Profits for compliant and non-compliant firms are given by Eqs. (4) and (12). By solving the profit maximization problem of the firm for any value of $\alpha _i$, it can be shown that firm level harvest is:

$$\begin{aligned} y_i^{*} = \left\{ \begin{array}{ll} \min \left( \frac{pX}{\alpha _i}, q\right) &{}\quad \text {for} \; \alpha _i\ge \hat{\alpha } \\ \frac{X}{\alpha _i} \left( p - \gamma f\right) &{}\quad \text {for} \; \alpha _i < \hat{\alpha } , \end{array} \right. \end{aligned}$$

(13)

where $\hat{\alpha }$ is the value of the firm-specific cost parameter $\alpha$ for which a firm would be indifferent between compliance and non-compliance.

1.4 Self-Report Based Enforcement System

To calculate aggregate harvest as a function of the self-reporting rebate when firms choose strategies A and B, we start out by analyzing optimal firm-level behavior. By maximizing the profit functions given above, we find that optimal catches are as follows:

$$\begin{aligned} y_a^*&= \frac{X}{\alpha }\left( p - rf\right) \end{aligned}$$

(14)

$$\begin{aligned} y_{b1}^*&= \frac{X}{\alpha }\left( p - \gamma _1 f\right) \end{aligned}$$

(15)

$$\begin{aligned} y_{b2}^*&= q , \end{aligned}$$

(16)

where subscripts $a$, $b1$, and $b2$ denote a firm choosing strategy A (in group 1), a firm choosing strategy B currently in group 1, and a firm choosing strategy B currently in group 2, respectively. By substituting catch response functions from Eqs. (14–16) into Eq. (3) and adjusting for the long-run shares of strategy B firms that are in groups 1 and 2, an expression for aggregate catch can be found.

We can now calculate the value of $\alpha$ for which a firm is indifferent between strategies A and B, which we denote $\alpha _0$. Strategy B, is relatively more attractive to more productive firms (low $\alpha _i$) because their gains from not self-reporting excess catches in group 1 are greater than for less productive firms (with high $\alpha _i$). Thus, if some firms prefer strategy B to strategy A it must be firms with low values of $\alpha _i$.

We now derive the value of $\alpha$ that makes a firm indifferent between strategies A and B, which we denote $\alpha _0$. The present value of all future payoffs for a firm following strategy A is:

$$\begin{aligned} EV_a = \sum \limits _{t=0}^\infty \beta ^t \pi _a^*\left( \alpha _i, X\right) , \end{aligned}$$

(17)

which can be rewritten:

$$\begin{aligned} EV_a = \frac{\pi _a^*\left( \alpha _i, X\right) }{1-\beta }. \end{aligned}$$

(18)

Correspondingly, the expected present value of all future payoffs for a firm following strategy B is:

$$\begin{aligned} EV_b = \sum \limits _{t=0}^\infty \beta ^t \pi _b^*\left( \alpha _i, X\right) . \end{aligned}$$

(19)

This can be rewritten as follows:³³

$$\begin{aligned} EV_b&= \pi _{b1}^*\left( \alpha _i, X\right) + (1-\gamma _1) \beta EV_b + \gamma _1 \left( \sum \limits _{t=0}^u \beta ^t \pi _{b2}^*\left( \alpha _i, X\right) + \beta ^{u+1} EV_b \right) \nonumber \\ EV_b&= \frac{\pi _{b1}^*\left( \alpha _i, X\right) + \gamma _1 \left( \sum _{t=0}^u \beta ^t \pi _{b2}^*\left( \alpha _i, X\right) \right) }{1-(1-\gamma _1) \beta - \gamma _1 \beta ^{u+1}}. \end{aligned}$$

(20)

The value of $\alpha _i$ that separates firms choosing strategy A from firms choosing strategy B can be identified by equating the present values of the two strategies ($EV_a = EV_b$) and is denoted $\alpha _0$. We substitute in for the maximized profit functions, $\pi _{a}^* = \frac{X}{2\alpha } \left( p - rf\right) ^2 + r f q$ and $\pi _{b1}^* = \frac{X}{2\alpha } \left( p - \gamma _1 f\right) ^2 + \gamma _1 fq$, and obtain:

$$\begin{aligned} \frac{\frac{X}{2\alpha _0} \left( p - rf\right) ^2 + r f q}{1-\beta } = \frac{\frac{X}{2\alpha _0} \left( p - \gamma _1 f\right) ^2 + \gamma _1 f q + \gamma _1 \sum _{t=0}^u \beta ^t \left( pq - \frac{\alpha _0 q^2}{2X}\right) }{1-(1-\gamma _1) \beta - \gamma \beta ^{u+1}} \end{aligned}$$

(21)

Rearranging the expression yields the following second order equation in $\alpha _0$:

$$\begin{aligned}&\alpha _0^2 (1\!-\!\beta ) \gamma _1 \sum \limits _{t=0}^u\left( \frac{\beta ^t q^2}{X} \right) \!-\! 2\alpha _0\left[ \gamma _1 (1\!-\!\beta ) \left( fq \!+\! \sum \limits _{t=0}^\infty \beta ^t pq\right) \!-\! rfq \left( 1\!-\!(1\!-\!\gamma _1) \beta \!-\! \gamma _1 \beta ^{u+1}\right) \right] \nonumber \\&\quad + X\left( p -rf\right) ^2 \left( 1-(1-\gamma _1) \beta - \gamma _1 \beta ^{u+1}\right) - X\left( p - \gamma _1 f\right) ^2 (1-\beta ) = 0 \end{aligned}$$

(22)

Solving Eq. (22) gives the following:

$$\begin{aligned} \alpha _0 = \frac{-B + \sqrt{B^2 - 4{ AD}}}{2A}, \end{aligned}$$

(23)

where $A$ , $B$ and $D$ are defined as follows:

$$\begin{aligned} A&= (1-\beta ) \gamma _1 \sum \limits _{t=0}^u \left( \frac{\beta ^t q^2}{X} \right) ,\\ B&= 2\left[ rfq \left( 1-(1-\gamma _1) \beta -\gamma _1 \beta ^{u+1}\right) - \gamma _1 (1-\beta ) \left( fq + \sum \limits _{t=0}^\infty \beta ^t pq\right) \right] ,\\ D&= X\left( p -rf\right) ^2 \left( 1-(1-\gamma _1) \beta - \gamma _1 \beta ^{u+1}\right) - X\left( p - \gamma _1 f\right) ^2 (1-\beta ) . \end{aligned}$$

Firms with $\alpha _i \ge \alpha _0$ find it optimal to use strategy A, while firms with lower production costs ($\alpha _i$) choose strategy B.

Finally, we calculate the aggregate catch response function under the assumption that $u$ can be set high enough to ensure that all firms chose strategy A. This requires that $u$ is set high enough for the inequality $\underline{\alpha } \ge \alpha _0(u)$ to hold, where $\alpha _0(u)$ is given by Eq. (23). We use the reaction function of strategy A firms from Eq. (14). In addition we know the probability density function of the uniformly distributed variable $\alpha$, which is $\frac{1}{\bar{\alpha }-\underline{\alpha }}$ (for $\underline{\alpha } \le \alpha \le \hat{\alpha }$). Given that there is a continuum of firms, total catches can be expressed as:

$$\begin{aligned} Y = \frac{n X \left( p - rf\right) }{\bar{\alpha }-\underline{\alpha }} \ln \left( \frac{\bar{\alpha }}{\underline{\alpha }}\right) . \end{aligned}$$

(24)