Walden et al.: Measuring change in productivity of a fishery with the Bennet-Bowley indicator 
275 
in overall productivity. Specifically, we wish to exam- 
ine how the productivity of vessels entering, exiting, 
and continuing within the fishery taken together influ- 
ences the aggregate productivity measure. In order to 
accomplish both tasks, we use the BB indicator, which 
is a price-weighted arithmetic mean of the difference 
in the change in output quantities and input quanti- 
ties used by firms (i.e., vessels) (Fare et al., 2008; Balk, 
2010). First, we show how to derive the BB indicator, 
and then how it can be decomposed into 3 components: 
1) the productivity of exiting vessels, 2) the produc- 
tivity of entering vessels, and 3) the productivity of 
continuing vessels within the fishery. This decomposi- 
tion will allow us to assess the contribution of each 
group to overall productivity. A similar approach was 
used to assess productivity gains in the mid-Atlantic 
individual transferable quota (ITQ) fishery for the At- 
lantic surfclam (Spisula solidissima) and ocean quahog 
(Arctica islandica) over a 30-year time period by using 
the Fare-Primont index (Fare et al., 2015). However, 
that approach required weighting individual productiv- 
ity scores by input distance functions. The BB indicator 
differs because it requires no weighting of individual 
productivity measures, and it can be constructed in 
spreadsheets. Therefore, it is easier than the Fare-Pri- 
mont approach for constructing the overall indicator. 
After examining the influence of entering and exiting 
vessels, we then extend the analysis in a different di- 
rection and use the additive nature of the BB indicator 
to determine how the composition of outputs produced 
and inputs used have influenced productivity change. 
The additive nature of the BB indicator allows us to 
see how landings mix and how changing input use 
have influenced productivity change. 
In terms of notation, let x 1 e 91$! be an input vector 
at time x and let y 1 e 91^ be an output vector, t=t,t+ 1. 
Let the corresponding prices be w 1 e 91^ and p 1 e 91^. 
The BB indicator takes the following form: 
which is a price-weighted difference between output 
change y t+ l - and input change x t+ l - x l . The weights 
used, which are the terms in the square brackets are 
formed by using directional vectors (g x ,g y ), g x e 9^ and 
g y e 9t^. Values need to be chosen for these vectors, and 
one possible choice is to set the directional vectors (g „ 
g y ) equal to the observed input and output levels. Do- 
ing so makes the denominators in the bracketed term 
equal to the sum of total revenue and total cost. In- 
stead, we set the value of the directional vectors equal 
to (1,1), which restricts the sum of the weights found in 
Equation 1 to equal one, and is consistent with share- 
valued weights. 
Note that the weights include both outputs and in- 
put prices — a consequence of the fact that the indicator 
is derived from profit maximization. A useful property 
of the BB indicator is additivity, i.e., that the vessel- 
level BB indicators can be added together and it will be 
equivalent to the industry-level calculated BB indica- 
tor. We assume that each group and each unit member 
face the same prices which gives us our desired decom- 
position, namely the ability to group our fleet into 3 
different sets of entering (N), continuing (C), and exit- 
ing (E) vessels, and the sum of the indicator for each 
group will equal the total indicator: 
££ t t+1 = (£B C )£ +1 + (B£ n )‘ +1 +(££ e )£ +1 . (3) 
To illustrate how the decomposition works, for new 
units the indicator equals 
(BB N )£ +1 
1 P 1 
2|p t g y + iu t g x 
+ 
P t+1 
1 
w t 
2 
P t gy+W t g x 
(4) 
Because these “new” units did not exist in period t, 
their inputs and outputs are zero at t. For the exiting 
units, inputs and outputs are zero in period t + 1 and 
therefore the indicator equals 
(BB e )[ +1 = - 
P^y + W l g x P^gy + W M g X 
P l g + W t g x P^gy + W M g x 
-Zyl] 
k=l J 
-14 
(5) 
Adjusting for a change in biomass with a volume 
indicator 
Fishing vessels produce landed fish that are extracted 
from a stock, and changes in productivity between 2 
periods are linked to the changes in fish stocks. For ex- 
ample, if a fish stock declines between years, a fishing 
vessel may still be able to maintain the same level of 
landings as those of the prior year by increasing effort, 
which means a greater use of inputs. Consequently, a 
productivity indicator such as the BB indicator would 
decline between years because the quantity of outputs 
would stay the same, whereas the quantity of inputs 
would increase. The relationship between productiv- 
ity change and biomass change has been recognized 
for some time now. Failing to account for a change in 
biomass results in a measure of productivity that has 
been called “biased” in the past (Squires, 1992). More 
recent studies have used the terms “biomass adjusted” 
and “biomass unadjusted” productivity change (Walden 
