Catch More to Catch Less: Estimating Timing Choice as Dynamic Bycatch Avoidance Behavior

Abstract

We model harvesters’ temporal participation behavior in a fishery with individual quotas for both a target and bycatch species. Harvesters make participation decisions given time-varying characteristics of the fishery (e.g., catch rates, price, and bycatch rates) and outside opportunities (e.g., other fisheries). A harvester’s problem is seasonally dynamic under the individual quota scheme because quota acts as an intertemporal budget constraint. We construct a theoretical model to describe how the shadow value of individual quota plays a role in a harvester’s decision and propose an empirical model that captures the dynamic effect of the seasonal quota usage. Our study finds support for the existence of dynamic bycatch avoidance: harvesters use the security provided by quota allocations to reduce harvesting around periods of high bycatch. Our policy simulation demonstrates that opening the season earlier could reduce bycatch while the main target catch is maintained due to temporal shift of quota usage.

Appendix

1.1 A1. Derivation of the Participation Index

The participation index for harvester i (Eq. 4) follows from the necessary first-order condition for the following constrained maximization problem:

$$\begin{aligned} \begin{array}{rl} \max \limits _{d_{it}} &{} V = \int ^{T}_0 [d_{it}(p_{1t}q_{1t} -\gamma b_{t}q_{1t}) + (1-d_{it})p_{2}q_{2t}-c]dt \\ \text {subject to} &{} \int _{0}^{T}d_{it}q_{1t}dt \le Q_{1i} \\ &{} \int _{0}^{T}d_{it}b_{t}q_{1t}dt \le Q_{bi} \\ &{} 0 \le d_{it} \le 1 \; \forall t. \end{array} \end{aligned}$$

The corresponding Lagrange function for the constrained maximization problem above is (including all inequality constraints):

$$\begin{aligned} {\mathcal {L}}= & {} V + \lambda _{1i}[Q_{1i} - \int ^{T}_{0} d_{it}q_{1t}dt] +\lambda _{bi}[Q_{bi} - \int ^{T}_{0}d_{it}b_{t}q_{1t}dt] \\&+ \int ^{T}_0\eta _{1it}d_{it}dt +\int ^{T}_0\eta _{2it}(1-d_{it})dt, \end{aligned}$$

where $\lambda _{1i}$, $\lambda _{bi}$, $\eta _{1it}$ and $\eta _{2it}$ are Lagrange multipliers corresponding to the target species quota constraint, the bycatch species quota constraint, the lower-bound constraint on $d_{it}$, and the upper-bound constraint on $d_{it}$, respectively. The solution to the constrained maximization problem can be characterized by the following necessary first-order conditions:

$$\begin{aligned}&\frac{\partial {\mathcal {L}}}{\partial d_{it}} = (p_{1t}q_{1t}-\gamma b_{t}q_{1t})-p_{2}q_{2t} - \lambda _{1i}q_{1t} - \lambda _{bi}b_{t}q_{1t} + \eta _{1it} - \eta _{2it}=0 \ \forall t \end{aligned}$$

(A1)

$$\begin{aligned}&\begin{aligned} \lambda _{1i}\left[ Q_{1i}-\int _{0}^{T}d_{it}q_{1t}dt\right]&= 0 \\ \lambda _{bi}\left[ Q_{bi}-\int _{0}^{T}d_{it}b_{t}q_{1t}dt\right]&= 0 \\ \eta _{1it}d_{it}&= 0 \ \forall t \\ \eta _{2it}(1-d_{it})&= 0 \ \forall t \\ \lambda _{1i},\lambda _{bi},\eta _{1it},\eta _{2it}&\ge 0 \ \forall t \end{aligned} \end{aligned}$$

(A2)

The participation index is derived by defining $Y_{it} \equiv \eta _{2it} - \eta _{1it}$ in Eq. (A1) and solving for $Y_{it}$:

$$\begin{aligned} Y_{it} = [p_{1t} - \lambda _{1i} - (\gamma + \lambda _{bi})b_{t}]q_{1t} - p_{2}q_{2t}. \end{aligned}$$

Intuitively, if the participation index is positive (i.e., the net benefits of fishing are higher in Fishery 1 than Fishery 2), then all effort is allocated to Fishery 1. Conversely, if the participation index is negative (i.e., the net benefits of fishing are higher in Fishery 2 than Fishery 1), then all effort is allocated to Fishery 2.

To see this formally, note that it is not possible for both the upper-bound and lower-bound constraints on $d_{it}$ to be binding simultaneously. Thus, it must be that:

• (1) $\eta _{1it} > 0$ and $\eta _{2it} = 0 \implies d_{it} = 0$,

• (2) $\eta _{1it} = 0$ and $\eta _{2it} > 0 \implies d_{it} = 1$, or

• (3) $\eta _{1it} = \eta _{2it} = 0 \implies 0 \le d_{it} \le 1$.

Case 1 simply says that if $Y_{it} \equiv \eta _{2it} - \eta _{1it} < 0$, then all fishing effort is allocated to Fishery 2 ($d_{it} = 0$). Conversely, Case 2 says that if $Y_{it} \equiv \eta _{2it} - \eta _{1it} > 0$, then all fishing effort is allocated to Fishery 1 ($d_{it} = 1$). Finally, Case 3 says that if $Y_{it} \equiv \eta _{2it} - \eta _{1it} = 0$, then a harvester is indifferent between the two fisheries and can allocate any proportion of effort between the two fisheries ($0 \le d_{it} \le 1$). For simplicity, we rule out this ambiguous case by assuming $d_{it} = I\{Y_{it} \ge 0\}$, meaning that the harvester would allocate all effort to Fishery 1 if they are indifferent between the two fisheries. In practice, this occurrence is rare and has no bearing on our empirical application.

1.2 A2. Derivations of Total Derivatives

In this section, we provide the full derivations of the total derivatives described in the model section.

As shown in the Eq. (5), the total derivative of the participation index with respect to bycatch rate is decomposed into two parts.

$$\begin{aligned} \frac{\text{d}Y_{it}}{\text{d}b_{t}} = \frac{\partial Y_{it}}{\partial b_{t}} + \frac{\partial Y_{it}}{\partial \lambda _{1i}}\frac{\partial \lambda _{1i}}{\partial b_{t}}I\{\lambda _{1i}> 0\} + \frac{\partial Y_{it}}{\partial \lambda _{bi}}\frac{\partial \lambda _{bi}}{\partial b_{t}}I\{\lambda _{bi} > 0\}. \end{aligned}$$

(A3)

The first term is the direct effect of the bycatch rate on participation, and is derived simply by taking the partial derivative of $Y_{it}$ in Eq. (4) with respect to $b_{t}$. The second and third terms are the indirect (or dynamic) effects of the bycatch rate on participation through its influence on the shadow values of quota. To derive these effects, we invoke the implicit function theorem to obtain the partial derivative of the shadow values with respect to the bycatch rate. Recall that shadow values are determined by the participation index (Eq. 4) in combination with the quota constraint conditions:

$$\begin{aligned} \begin{aligned} G_{1}(b_{t},\lambda _{1i})&= Q_{1i} - \int ^{T}_{0} d_{it}q_{1t}dt \ge 0 \\ G_{b}(b_{t},\lambda _{bi})&= Q_{bi} - \int ^{T}_{0} d_{it}b_{t}q_{1t}dt \ge 0. \end{aligned} \end{aligned}$$

(A4)

and the equality holds when the constraints are binding, implying that $\lambda _{1i} >0$ and $\lambda _{bi} >0$, respectively. Suppose the constraint of main target species quota is binding. The derivative of the shadow value for target species quota with respect to the bycatch rate is

$$\begin{aligned} \begin{aligned} \frac{\partial \lambda _{1i}}{\partial b_t}&= -\frac{\frac{\partial G_{1}}{\partial b_t}}{\frac{\partial G_{1}}{\partial \lambda _{1i}}} \\&= -\frac{-\frac{\partial d_{it}}{\partial Y_{it}}\frac{\partial Y_{it}}{\partial b_{t}} q_{1t}}{-\int ^{T}_{0} \frac{\partial d_{is}}{\partial Y_{is}}\frac{\partial Y_{is}}{\partial \lambda _{1i}}q_{1s}ds} \\&= -\frac{(\gamma + \lambda _{bi})\frac{\partial d_{it}}{\partial Y_{it}}q^{2}_{1t}}{\int ^{T}_{0} \frac{\partial d_{is}}{\partial Y_{is}}q^{2}_{1s}ds} \le 0. \end{aligned} \end{aligned}$$

(A5)

where the function $G_{1}$ is the binding constraint of the target species quota, which is defined when $\lambda _{1i}>0$ (i.e., when the quota constraint is binding). Recall that $d_{it}$ is a function of $Y_{it}$, which in turn is a function of $b_{t}$ and $\lambda _{1i}$. Hence, the derivative $\frac{\partial \lambda _{1i}}{\partial b_t}$ is defined. Notice that changes in the bycatch rate in period t only influence the contemporaneous participation index but changes in the shadow value of the quota constraint change the participation index in all periods. Combined with the effect of the shadow value on contemporaneous participation, $\frac{\partial Y_{it}}{\partial \lambda _{1i}} = -q_{1t}$, we have the following expression for the second term in Eq. (A3),

$$\begin{aligned} \frac{\partial Y_{it}}{\partial \lambda _{1i}}\frac{\partial \lambda _{1i}}{\partial b_{t}} = q_{1t}\frac{(\gamma + \lambda _{bi})\frac{\partial d_{it}}{\partial Y_{it}}q^{2}_{1t}}{\int ^{T}_{0} \frac{\partial d_{is}}{\partial Y_{is}}q^{2}_{1s}ds} \ge 0, \end{aligned}$$

(A6)

which is unambiguously positive. Thus, the dynamic effect of the bycatch rate through the shadow value of the target species quota counters, but does not completely offset, the direct effect of the bycatch rate on participation.

We can follow a similar procedure for deriving the third term in Eq. (A3). The derivative of the shadow value for bycatch species quota with respect to the bycatch rate is

$$\begin{aligned} \begin{aligned} \frac{\partial \lambda _{bi}}{\partial b_t}&= -\frac{\frac{\partial G_{b}}{\partial b_t}}{\frac{\partial G_{b}}{\partial \lambda _{bi}}} \\&= -\frac{-(d_{it} + \frac{\partial d_{it}}{\partial Y_{it}}\frac{\partial Y_{it}}{\partial b_{t}}b_{t})q_{1t}}{-\int ^{T}_{0} \frac{\partial d_{is}}{\partial Y_{is}}\frac{\partial Y_{is}}{\partial \lambda _{bi}}b_{s}q_{1s}ds} \\&= -\frac{-(d_{it} + \frac{\partial d_{it}}{\partial Y_{it}}[-(\gamma + \lambda _{bi})q_{1t}]b_{t})q_{1t}}{-\int ^{T}_{0} \frac{\partial d_{is}}{\partial Y_{is}}(-b_{s}q_{1s})b_{s}q_{1s}ds} \\&= \frac{(d_{it} - (\gamma + \lambda _{bi}) \frac{\partial d_{it}}{\partial Y_{it}}b_{t}q_{1t})q_{1t}}{\int ^{T}_{0} \frac{\partial d_{is}}{\partial Y_{is}}b^{2}_{s}q^{2}_{1s}ds}, \end{aligned} \end{aligned}$$

(A7)

the sign of which is ambiguous and depends on the value of the participation index $Y_{it}$. For example, if $Y_{it} > 0$ so that a vessel is already participating in Fishery 1, then $d_{it} = 1$ and $\frac{\partial d_{it}}{\partial Y_{it}} = 0$, which implies that $\frac{\partial \lambda _{bi}}{\partial b_t} > 0$. Intuitively, the shadow value of bycatch quota will increase with the bycatch rate so long as a vessel derives a benefit from having more bycatch quota in terms of increased target species catch in Fishery 1. Conversely, if $Y_{it} < 0$ so that a vessel is participating in Fishery 2, then $d_{it} = 0$ and $\frac{\partial d_{it}}{\partial Y_{it}} = 0$, which implies no impact on the shadow value because $\frac{\partial \lambda _{bi}}{\partial b_t} = 0$. In this case, a vessel derives no value from additional bycatch quota since no bycatch is encountered in Fishery 2. The only case in which the shadow value of bycatch quota will decrease with the bycatch rate is if the increased cost of bycatch is large enough to push a vessel from Fishery 1 into Fishery 2. In this case, $Y_{it} = 0$, $d_{it} = 1$, and $\frac{\partial d_{it}}{\partial Y_{it}} = 1$, which implies that $\frac{\partial \lambda _{bi}}{\partial b_t} < 0$ if and only if $1 > (\gamma + \lambda _{bi})b_{t}q_{1t}$. Combined with the effect of the shadow value on contemporaneous participation, $\frac{\partial Y_{it}}{\partial \lambda _{bi}} = -b_{t}q_{1t}$, we have the following expression for the third term in Eq. (A3):

$$\begin{aligned} \frac{\partial Y_{it}}{\partial \lambda _{bi}}\frac{\partial \lambda _{bi}}{\partial b_{t}} = -b_{t}q_{1t}\frac{(d_{it} - (\gamma + \lambda _{bi}) \frac{\partial d_{it}}{\partial Y_{it}}b_{t}q_{1t})q_{1t}}{\int ^{T}_{0} \frac{\partial d_{is}}{\partial Y_{is}}b^{2}_{s}q^{2}_{1s}ds}. \end{aligned}$$

(A8)

Hence, the total derivative of the participation index with respect to the bycatch rate is expressed as the Eq. (6).

The total derivatives of the participation index with respect to other variables ($\frac{\partial Y_{it}}{\partial q_{1t}}$, $\frac{\partial Y_{it}}{\partial Q_{1i}}$,$\frac{\partial Y_{it}}{\partial Q_{bi}}$) can be derived in a similar manner. We provide the partial derivatives that are necessary for the derivations in the next appendix section.

1.3 A3. Partial Derivatives

The partial derivative of shadow values with respect to the catch rate of main target species.

$$\begin{aligned} \begin{aligned} \frac{\partial \lambda _{1i}}{\partial q_{1t}}&= -\frac{\frac{\partial G_1}{\partial q_{1t}}}{\frac{\partial G_{1}}{\partial \lambda _{1i}}} \\&= -\frac{-\left( \frac{\partial d_{it}}{\partial Y_{it}}\cdot \frac{\partial Y_{it}}{\partial q_{1t}} + d_{it}\right) }{-\int ^{T}_{0}\frac{\partial d_{is}}{\partial Y_{is}}\frac{\partial Y_{is}}{\partial \lambda _{1i}}q_{1s}ds} \\&= \frac{\left\{ \frac{\text{d}d_{it}}{\text{d}Y_{it}}\left[ p_{1t} - \lambda _{1i} - (\gamma + \lambda _{bi})b_{t} \right] + d_{t} \right\} }{\int ^{T}_{0}\frac{\text{d}d_{is}}{\text{d}Y_{is}}q^{2}_{1s}ds} \end{aligned} \end{aligned}$$

(A9)

The sign of the effect depends on the sign of the net benefit per unit catch of the main target.

$$\begin{aligned} \begin{aligned} \frac{\partial \lambda _{bi}}{\partial q_{1t}}&= -\frac{\frac{\partial G_b}{\partial q_{1t}}}{\frac{\partial G_{b}}{\partial \lambda _{bi}}} \\&= -\frac{-\left( \frac{\partial d_{it}}{\partial Y_{it}}\cdot \frac{\partial Y_{it}}{\partial q_{1t}} + d_{it}b_{t}\right) }{-\int ^{T}_{0}\frac{\partial d_{is}}{\partial Y_{is}}\frac{\partial Y_{is}}{\partial \lambda _{bi}}b_{s}q_{1s}ds} \\&= \frac{\left\{ \frac{\text{d}d_{it}}{\text{d}Y_{it}}\left[ p_{1t} - \lambda _{1i} - (\gamma + \lambda _{bi})b_{t} \right] + d_{t}b_{t} \right\} }{\int ^{T}_{0}\frac{\text{d}d_{is}}{\text{d}Y_{is}}b_{s}^{2}q^{2}_{1s}ds} \end{aligned} \end{aligned}$$

(A10)

The sign of the effect depends on the sign of the net benefit per unit catch of the main target.

The partial derivative of shadow values with respect to the main target quota.

$$\begin{aligned}&\begin{aligned} \frac{\partial \lambda _{1i}}{\partial Q_{1i}}&= -\frac{\frac{\partial G_1}{\partial Q_{1i}}}{\frac{\partial G_{1}}{\partial \lambda _{1i}}} \\&= -\frac{1}{-\int ^{T}_{0}\frac{\partial d_{is}}{\partial Y_{is}}\frac{\partial Y_{is}}{\partial \lambda _{1i}}q_{1s}ds} \\&= -\frac{1}{-\int ^{T}_{0}\frac{\text{d}d_{is}}{\text{d}Y_{is}}(-q_{1s})q_{1s}ds} \\&= -\frac{1}{\int ^{T}_{0}\frac{\text{d}d_{is}}{\text{d}Y_{is}}q^{2}_{1s}ds} < 0 \end{aligned} \end{aligned}$$

(A11)

$$\begin{aligned}&\begin{aligned} \frac{\partial \lambda _{bi}}{\partial Q_{1i}}&= -\frac{\frac{\partial G_b}{\partial Q_{1i}}}{\frac{\partial G_{b}}{\partial \lambda _{bi}}} \\&= -\frac{0}{-\int ^{T}_{0}\frac{\partial d_{is}}{\partial Y_{is}}\frac{\partial Y_{is}}{\partial \lambda _{1i}}q_{1s}ds} \\&= -\frac{0}{-\int ^{T}_{0}\frac{\text{d}d_{is}}{\text{d}Y_{is}}(-q_{1s})q_{1s}ds} \\&= -\frac{0}{\int ^{T}_{0}\frac{\text{d}d_{is}}{\text{d}Y_{is}}q^{2}_{1s}ds} = 0 \end{aligned} \end{aligned}$$

(A12)

The partial derivative of shadow values with respect to the bycatch target quota.

$$\begin{aligned}&\begin{aligned} \frac{\partial \lambda _{1i}}{\partial Q_{bi}}&= -\frac{\frac{\partial G_1}{\partial Q_{bi}}}{\frac{\partial G_{1}}{\partial \lambda _{1i}}} \\&= -\frac{0}{-\int ^{T}_{0}\frac{\partial d_{is}}{\partial Y_{is}}\frac{\partial Y_{is}}{\partial \lambda _{1i}}q_{1s}ds} \\&= -\frac{0}{-\int ^{T}_{0}\frac{\text{d}d_{is}}{\text{d}Y_{is}}(-q_{1s})q_{1s}ds} \\&= -\frac{0}{\int ^{T}_{0}\frac{\text{d}d_{is}}{\text{d}Y_{is}}(b_{s}q_{1s})^{2}ds} = 0 \end{aligned} \end{aligned}$$

(A13)

$$\begin{aligned}&\begin{aligned} \frac{\partial \lambda _{bi}}{\partial Q_{bi}}&= -\frac{\frac{\partial G_b}{\partial Q_{bi}}}{\frac{\partial G_{b}}{\partial \lambda _{bi}}} \\&= -\frac{1}{-\int ^{T}_{0}\frac{\partial d_{is}}{\partial Y_{is}}\frac{\partial Y_{is}}{\partial \lambda _{1i}}q_{1s}ds} \\&= -\frac{1}{-\int ^{T}_{0}\frac{\text{d}d_{is}}{\text{d}Y_{is}}(-q_{1s})q_{1s}ds} \\&= -\frac{1}{\int ^{T}_{0}\frac{\text{d}d_{is}}{\text{d}Y_{is}}(b_{s}q_{1s})^{2}ds} < 0 \end{aligned} \end{aligned}$$

(A14)

1.4 A4. Modeling Expectations

In our estimation of Eq. (10), we employ proxies of expected revenues EREV and bycatch rates ECPR. To form proxies of weekly-level expectations of catch, we assume that harvesters know the distribution of seasonal catch and bycatch rates. There are two key aspects for formulating catch expectations in the fisheries literature: (1) common and private information, and (2) temporal and spatial resolution of information. While some studies assume that harvesters use only common information and utilize a rolling average or autoregressive moving average as a common expectation associated with fishing alternatives (e.g., Curtis and Hicks 2000; Curtis and McConnell 2004; Smith and Wilen 2003), recent work considers the role of private information to form individual expectations with fine resolution of data (Abbott and Wilen 2011). At the week level, however, idiosyncratic information may not play a large role in the participation choice; instead, prior knowledge about seasonality and the updated current season information would matter most. In addition, we aggregate fine-grained information to model weekly level decisions. Thus, we model catch expectations using weekly and annual trends, in addition to time invariant vessel effects.

We first estimate weekly standardized catch per unit effort (CPUE) and bycatch rates. To capture seasonal trends in the data, we estimate standardized catch per unit effort (haul-hour) and bycatch rate (Chinook–pollock ratio) for each week, assuming a log-normal and Poisson distribution, respectively, and the following specifications for the mean:

$$\begin{aligned}&\ln (PollCPUE_{it}) = \sum _t \delta _t DW_t + \sum _t \delta _t DY_t + \sum _i \delta _i DV_i \end{aligned}$$

(A15)

$$\begin{aligned}&\ln (Chin_{it}) = \sum _t \eta _t DW_t + \sum _t \eta _t DY_t + \sum _i \eta _i DV_i + \ln Poll_{it}, \end{aligned}$$

(A16)

where DW is a week dummy variable, DY is a year dummy, and DV is an individual vessel dummy. The weekly standardized CPUEs and bycatch rates are estimated as the vectors $\delta$ and $\eta$. We assume that harvesters base their beliefs on within-season trends of catch and bycatch rates that are smooth over a season. Hence, we apply a local regression method (LOESS) to the estimated weekly CPUEs and bycatch rates to obtain smooth seasonal trends. Given the assumption that vessels know the true distribution of catch, we use all periods and vessels in the sample to estimate the standardized CPUEs and bycatch rates. Harvesters’ expectations are assumed to be based on the seasonality which is formed at the fleet level and taken as exogenous for each vessel.

The weekly expected CPUEs of individual harvesters are formed using the estimated seasonal trend (common information) and the observed standard CPUE of the previous week (individual information). We assume that individuals form rational expectations based on those variables, regress the trend and 1-week lagged CPUE on the current CPUE, and use the fitted values as individual expectations. Table 5 shows the result of the estimated model of rational expectations. As expected, both of the common and individual information are important for the formation of the expectation.

Note that our measure of expected bycatch rates are the product of both intra-annual mixing of salmon and pollock and underlying bycatch avoidance decisions of the entire fleet (e.g., spatial avoidance). Hence, the expected bycatch rate in each period reflects the best practice of bycatch avoidance under existing measures. The expected bycatch uses information from the whole fleet; an individual harvester’s participation decisions are only a small contribution to this measure, so we believe the degree to which this measure is endogenous is small. We acknowledge that our measure of expected bycatch is not completely exogenous (i.e., natural mix of Chinook salmon and pollock), but the impact of endogeneity in terms of estimation bias is negligible.

Figure 6 shows the observed and expected pollock CPUE and Chinook–pollock ratio. As Panels A and B show, there are some large outliers in the observed data, but the weekly mean exhibits trends across a season. The pollock CPUE is relatively stable over the A season but decreases midway through the B season.The Chinook–pollock ratio starts high in the beginning of A season, reduces toward the end of the A season and beginning of the B season, and then increases again towards the end of the B season. These trends are largely captured by the predicted expectations, depicted by the solid lines in Panels C and D. Each individual harvester forms their expectation based on this common trend, as well as individual information based on the result of Table 5.

Table 5 Estimation results of the expected pollock CPUE and Chinook–pollock ratio

Fig. 6

Seasonal variation of Pollock CPUE and Chinook–pollock ratio, A observed pollock CPUE, B observed Chinook–pollock ratio, C expected pollock CPUE and D expected Chinook–pollock ratio. The grey lines in panel (C) and (D) indicate the in-season trends