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 timevarying 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.
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Notes
Number of appearance is reported on annual reports. e.g. Pollock Conservation Cooperative and High Sea Catchers’ cooperative join annual report, https://www.npfmc.org/wpcontent/PDFdocuments/catch_shares/CoopRpts2016/PCC_HSCC_AFA16.pdf.
For example, Birkenbach et al. (2020) includes discounting in their theoretical model for completeness, but not explicitly treat it in their empirical section. We exclude the discounting to keep the expression simple.
Note that a harvester is indifferent between the two fisheries when \(Y_{it} = 0\). In this case, \(\eta _{1it} = \eta _{2it} = 0\) and \(0 \le d_{it} \le 1\). For simplicity, we assume that a harvester would allocate all effort to Fishery 1 if indifferent. In practice, this is rare. We provide a full derivation of the necessary condition in Eq. (4) in Appendix A1.
A full derivation of the derivative is provided in the Appendix A2.
This is not an openaccess, because the fishery is not open to public. It is open in the sense that the catcherprocessor fleet can access without quota regulation.
Following the literature (e.g., Abbott and Wilen 2011), we estimate harvesters’ expectations outside of the fishery participation decision model. The formation and estimation of such expectations are discussed in detail in Appendix A4. We note that a potential problem with using estimates of expectations is that they likely contain measurement and prediction errors, which can lead to attenuation bias (assuming that these errors are not systematically related to the latent expectation). We also note that since we model expectations as a function of previous participation decisions, there is a possibility that our expectations are correlated with the unobserved component of indirect utility, resulting in endogeneity bias. However, we believe that such endogeneity bias is likely small since: (i) an individual harvester contributes only a small portion to fleetwide harvests; (ii) the stochastic nature of bycatch rates means that there is considerable exogenous variation in bycatch, conditional on participation decisions; and (iii) the inclusion of vessel fixed effects captures any endogeneity arising from unobserved factors that are vessel specific and constant over time. We thank an anonymous reviewer for pointing this out.
The actual inseason variations of exvessel or wholesale prices were not available.
Anecdotal evidence suggests that while limits on salmon bycatch do influence harvesting behavior, harvesters tolerate a higher level of bycatch for the greater value of mature roe.
We note that performing such policy simulations does not require identification of deep primitive structural parameters; rather, only combinations of structural parameters need to be identified, so long as they remain the same under the different policies we consider—i.e., they are policy invariant (Heckman 2010). Thus, an important assumption we make is that the parameters we identify in the indirect utility function (Eq. 10) are policy invariant. The performance of our outofsample predictions provides evidence that our participation model is capturing mechanisms that are relevant for conducting counterfactual policy simulations. We thank an anonymous reviewer for raising this issue.
The slight overprediction of the simulated participation under status quo may be due to prediction error of pollock catch: predicted catch is not exactly the same as the actual catch, and hence Quota Speed and the participation in the next week may have prediction error. Our focus is the difference between the status quo and the policy alternative.
The total seasonal catch of pollock seems to increase by a small amount. This is because catch predictions may exceed the quota in the last week of participation in the simulation process. In reality, there is no reason that the total pollock catch should increases since the individual quota is binding under the status quo.
In 2006, many nonChinook salmons were exceptionally caught in the early season, resulting in an increase of 647 nonChinook salmon caught as bycatch under the alternative policy (a relative increment of only 2.6%).
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Appendix
Appendix
1.1 A1. Derivation of the Participation Index
The participation index for harvester i (Eq. 4) follows from the necessary firstorder condition for the following constrained maximization problem:
The corresponding Lagrange function for the constrained maximization problem above is (including all inequality constraints):
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 lowerbound constraint on \(d_{it}\), and the upperbound constraint on \(d_{it}\), respectively. The solution to the constrained maximization problem can be characterized by the following necessary firstorder conditions:
The participation index is derived by defining \(Y_{it} \equiv \eta _{2it}  \eta _{1it}\) in Eq. (A1) and solving for \(Y_{it}\):
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 upperbound and lowerbound 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.
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:
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
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),
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
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):
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.
The sign of the effect depends on the sign of the net benefit per unit catch of the main target.
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.
The partial derivative of shadow values with respect to the bycatch target quota.
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 weeklylevel 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 finegrained 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 (haulhour) and bycatch rate (Chinook–pollock ratio) for each week, assuming a lognormal and Poisson distribution, respectively, and the following specifications for the mean:
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 withinseason 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 1week 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 intraannual 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.
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Abe, K., Anderson, C.M. & Reimer, M.N. Catch More to Catch Less: Estimating Timing Choice as Dynamic Bycatch Avoidance Behavior. Environ Resource Econ 82, 953–984 (2022). https://doi.org/10.1007/s10640022006996
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DOI: https://doi.org/10.1007/s10640022006996
Keywords
 Bycatch
 Dynamic avoidance
 Policy simulation
 Prohibited species catch
 Shadow value of individual quota
JEL Classification
 C61
 Q22
 Q28