# Combining bioeconomic modelling and valuation (1): The basics

This is the first of a series of blog posts focusing on bioeconomic modelling and how to combine it with environmental valuation.

This week I decided to develop my own bioeconomic model. I have to admit that bioeconomic modelling is outside my field of expertise, despite coming across with it during my masters. However, there is one particular opportunity that comes from being acquainted with these kind of models, that is, to merge them with environmental valuation.

Bioeconomic modelling is “traditionally used to derive optimal fish stock and harvest rates” (Armstrong et al., 2017).

Sometimes bioeconomic modelling combined with environmental valuation can prove to be very policy-relevant. For example, let us say that there is a species of rare fish that is harvested for consumption and has great value for fishermen. Concurrently, that species is also the reason why some visitors come to the region to go scuba diving. It goes without saying that scuba divers want to be able to see this fish species, and not spend an unnecessarily long amount of time underwater. Hence, the value for commercial fisheries and for the scuba diving industry depends on the stock of this one species of fish. It could be beneficial if both values related to the stock could be included when modelling how optimal harvesting should look like.

However, the value of the scuba diving industry does not necessarily have a market price. Especially if divers bring their own equipment and perform the activities without purchasing any services. In that case, environmental valuation is one option to find the value of scuba diving (per trip or per sighting of fish). It can then be included in the bioeconomic model as a function of the stock of fish.

There are a few studies that combine these optimal harvesting models with values from non-market valuation. Armstrong et al. (2017) is one of these studies that expands their bioeconomic fisheries model where they recognize that natural habitats provide non-use benefits. They apply their model to cold-water corals in Norway and analyze harvesting of the Northeast artic cod. They find that including non-use values increases optimal coral habitat by 25%, and decreases optimal fish stock by 7%.

Vondolia et al. (2020) develop a bioeconomic model including both fisheries management and several co-benefits provided by coastal cod and kelp forests in Norway. These co-benefits include carbon sequestration, for example.

This tutorial was a lifesaver to get started with developing a basic bioeconomic model. It is a document developed by Rolf Groeneveld available at his webpage. Not only does it explain all the necessary steps to create the model, it also introduces basic R language for the less experienced user. I will follow the tutorial while I explain the necessary steps to develop the model which uses fisheries as the example.

The exposition starts by recognizing that the stock (i.e. the available quantity at any point in time) depends on two things. The first one is the previous stock. In other words, the amount of knowledge I possess depends on how much knowledge I had the previous year. The second is the growth rate of the resource. The faster knowledge grows, the more knowledge I will have in the present compared with in the past. Hence, the stock (denoted by X) at any given point in time is given by:

$X_t = X_{t-1} + r*X_{t-1}$

$t$ is the subscript for time (years, months, weeks, etc) and $r$ represents the growth rate of the resource.

I can write this in R by typing:

X[2] <- X[1]+r*X[1]

Below I have specified that the stock size at time 2 is equal to the stock size in the previous period (t=1) plus the stock size in the previous period times the growth rate of the resource (r).

Of course, before running the expression above, you have to specify the missing variables. These are the growth rate (r) and the stock sizes at each point in time (X). I will use the same values as in the tutorial for the growth rate and other parameters:

 r <- 0.05
X <- c(rep(0, times = numberOfYears)) 

I will look at a longer time horizon (50 years):

numberOfYears <- 50

We also need to tell the code what the initial biomass/stock is, that is, the stock at time 1. It cannot be zero, otherwise that would mean there are no fishes in this ecosystem. Let us assume that the initial stock is 10 (tons of biomass for example).

X[1] <- 10

We now have all the necessary parameters to calculate the stock at different points in time. For now, let us pretend we are biologists looking at a pristine area where fishes are not under any human pressures. In other words, fishermen are not harvesting the fish. The stock grows every year, since fish reproduce in the ecosystem at the growth rate we have set previously (r=5%). Under no pressure, the fisheries will keep increasing until it reaches its carrying capacity. The carrying capacity, which we will represent with m, is the maximum population size of a species (in this case an undetermined species of fish) that can be sustained in that specific environment given all the resources that are available.[1]

Let us suppose that the maximum biomass of this species of fish (denoted by $m$) would be 70 (tons). In other words, the stock will not go beyong 70, since there is, for example, not enough food for that many tons of fish.

m <- 70

The stock function above needs to be adjusted, to what is called the Gordon-Schaefer growth function:

$X_t = X_{t-1} + r*X_{t-1}(1-\frac{X_{t-1}}{m})$

In R, we can type the Gordon-Schaefer function as a loop, defining for each time period the existing stock given all the defined parameters above:

for(t in 2: numberOfYears){
X[t] <- X [t-1]+rX[t-1](1-X[t-1]/m)
}

The result is the vector X (our stock vector) that defines for every time period how much stock there is. We can visualize the evolution of the stock over time by plotting the X vector:

plot(X)

We can see that over the 50 years, the stock grows steadily but it will not reach the 70 tons of carrying capacity, since it cannot grow that fast.

Like I said, this model would suffice if the fishery was not under any kind of human pressure. As we know, fishing is an important economic activity which has impacts on the amount of fish available in a specific ecosystem. In the next post, I will introducing fish harvesting into the picture.

References:

Armstrong, C. W., Kahui, V., Vondolia, G. K., Aanesen, M., & Czajkowski, M. (2017). Use and non-use values in an applied bioeconomic model of fisheries and habitat connections. Marine Resource Economics32(4), 351-369.