2. Open Access

Question 1

Write down the sensible growth model

Answer 1

G(S) = aS (1 – S/K)

Question 2

Reasons for surplus growth in the growth model?

Answer 2

1. Fewer fish, more food available for each
2. Fishing means fewer old, slow-growing fish (fishing target older and larger fish)
3. Fewer old fish could mean less predation on young fish

Question 3

If S < S_msy.

Answer 3

• Biological overexploitation
• But do not need to be economically overexploited
• Might be economically defensible

Question 4

What is the simple “classical” fishery model?

Answer 4

Y=EqS
1. E = fishing effort. The activity of removing fish from the sea
2. One unit of effort removes a certain fraction of the fish stock (S)
3. Y = catch of fish
4. q = constant coefficient related to how we measure effort and stock (catchability, or availability coefficient)

Question 5

Show a simple example of how biological overexploitation can be justified

Answer 5

1. Surpluss growth function: G(S)=aS(1-S/K). Also, lets say that a 0.5, K=1, S_msy=0.5
2. Draw graph. Overexploited to the left, underexploited to the right, fully exploited at S_msy in the middle.
3. (a) G(0.5) = 0.5*0.5 * (1 - 0.5/1) = 0.125
4. We now reduce the stock from 0.5 to 0.45. This gives an immediate gain by p=∆S=0.05, where p is the price of fish net cost
   • 0.05 increase in captures
   • But smaller stock is less productive
5. (b) G(0.45) = 0.5 * 0.45 * 0.55 = 0.12375
6. Annual loss is (a)-(b) 0.125 – 0.12375 = 0.00125
7. The present value of the loss in perpetuity is 0.00125/r. With 5% discount rate, we get 0.025
8. 0.05 > 0.025 (with 5% discount rate). The value of the one-tme gain is bigger than the perpetuity loss
   • Makes sense to take a smaller sustainable yield than the maximum.

Question 6

Find sustainable stock level and sustainable yield as a function of effort

Answer 6

The logistic growth function:
(1) aS(1-S/K)

Catch of fish:
(2) Y=qES

Put (1) and (2) equal and we get the sustainable stock level as a function of effort:
(3) S_sus=K(1-Eq/a)

Insert this stock level into (2) and we get the sustainable yield as a function of effort:
(1) Y_sus = E – βE^2
 = qK
β = q^2K/a

Question 7

Find sustainable stock level and sustainable yield as a function of effort

Answer 7

The logistic growth function:
(1) aS(1-S/K)
Catch of fish:
(2) Y=qES
We set the catch of fish (2) equal to the growth of fish (1):
(3) aS(1-S/K) = qES
Solving for S, we get the sustainable fish stock level as a function of fishing effort:
(4) S_sus=K(1-Eq/a)
Next, we want to find the sustainable yield. We know that alpha and beta is:
(5) Β = q^2K/a
(6)  = qk
Formula for Y_sus (sustainable yield) is now:
(7) Y_sus=qEK(1-Eq/a)
Insert this stock level into (2) and we get the sustainable yield as a function of effort:
(7) Y_sus = E – βE^2
 = qK
β = q^2K/a

Question 8

What are the assumptions?

Answer 8

1. Constant cost per unit of effort
   * Some fishermen are more clever than other, or have better equipment and earn profits (skill or equipment rents)
   * Implies a rising curve of cost per unit of effort
   * The marginal fisherman still breaks even (unit cost = average product > marginal product), but we still have overexplotation under open access)
2. Quantity-dependent price of fish, p=f(Y)
   * The sustainable yield curve can have two peaks, and we can get three equilibria with open access, but there will still be overexploitation (MP < c)
3. A different production function, such as
   Y = ES^b
   0 ≤ b ≤ 1
   q = 1

• b < 1 could be due to how fish change their distribution in the sea as the stock dimishes. Open access would still result in overexploitation and possibly extinction
• Catch per unit of effort: Y/E = ES^b/E = S^b
• If b < 1, Y/E falls less quickly than the stock
• If b = 0, fish stocks would go extinct under open access, pY/E > c always and there is always an incentive to expand fishing effort
• For demersal fish, b probably close to 1, for pelagic fish which move in shoals (herring, mackerel, etc., close to 0).

Question 9

What are the consequences of open access?

Answer 9

• Leads to economic overexploitation
• Leads to extinction of fish stocks if cost per unit of fish caught is independent of stock size (think buffalo)

a) So fishermen have incentive to max short-term, neglecting long-term

Question 10

When do we have biological overexploitation?

Answer 10

When the fish stock is S < S_msy

Question 11

Q: What is the relation between biological and economical overexploitation?

Answer 11

a) economic overexploitation typically means biological overexploitation
b) but biological overexploitation does not necessarily mean economical overexploitation (can be justified)

Question 12

Can biological overexploitation be justified?

Answer 12

Can be justified economically, if the cost per unit of fish caught is insensitive to the stock size. (some loss of future sustainable yield can be traded off against unsustainable gain)

a) cost per unit of gain is INSENSITIVE to stock size. Expenses remain constant

Question 13

Q: Explain the difference between a open-access fishery and a regulated fishery (text)

Answer 13

Open-access:
* Available to all (who, how much, when).
* Act in self-interest, and behave contrary to the common good by depleting the shared resource.

Open, economically
* Continue to fish as long revenue > cost.
* May lead to NOT sustainable level, both biologically and economically
* Effort could be greater than sustainable yield, which could lead to overfishing

Regulated:
* Rules and regulations
* Limits on allowable catch, restrictions on gear or techniques, limitations on fishing season
* Aim: Sustainability and NOT deplete fish stocks

Regulated, economically:
* Aim is to achieve maximum sustainable yield (highest yield that can be sustainably fished from the stock year after year)
* This level is achieved by controlling the fishing effort such that the marginal productivity, the value contributed by the last unit of effort, is equal to the cost per unit of effort.
* Prevents overexploitation of the fish stock and ensures that the fishing activities are economically sustainable.

Question 14

Q: Explain the difference between a open-access fishery and a regulated fishery (show graphs)

Answer 14

Open Access Explanation:
* Individuals self-interest

Figure 4.3 Upper Panel:
* Sustainable yield: represents the amount of fish that can be harvested sustainably at different levels of effort
* Cost lines: indicate the total cost incurred for different fishing efforts
* E1^* and E2^*: represent open-access equilibria, where the total revenues from the fishing match the total costs

You can see that, paradoxically, lower costs lead to more fishing effort and less sustainable yield, indicative of overfishing in an open-access scenario. The lower panel of Figure 4.3 helps clarify this.
* The downward-sloping AP (Average Productivity) and MP (Marginal Productivity) lines reflect the diminishing returns in fishing productivity with increasing fishing effort.
* Under open-access conditions, equilibrium occurs when AP equals the cost per unit of effort (at points E1^* and E2^*), meaning the last unit of effort is not paying for itself—it contributes less to the total value than it costs.

Regulated Fishery:

Figure 4.3 Upper Panel:
* The points E1^0 and E2^0 represent the economically optimal level of fishing effort, where the marginal productivity line intersects with the cost per unit of effort line.
* This implies less effort than under open access, and the effort would not exceed the effort producing the maximum sustainable yield.

Figure 4.3 Lower Panel:
* The optimal effort levels E1^0 and E2^0 occur when the Marginal Productivity (MP) equals the cost per unit of effort.
* Here, each unit of effort or boat contributes a net value to the fishery equal to its cost, maximizing the difference between revenues and costs—known as the resource rent or fishing rent.

Question 15

what is effort?

Answer 15

Can be a lot and can be measured in different ways:
1. Hours of trawling
2. Size of boats
3. Number of hooks applies
4. Etc.

Question 16

Q: In the logistic growth function, if S < K…

Answer 16

The surplus growth is positive.

Question 17

Q: What are the arguments for taking the sustainable yield in the logistic growth function, and why are they wrong?

Answer 17

The arguments are:
a) Less risk of fish population going extinct
b) Would be cheaper

But there is a tradeoff of immediate gain and long-term loss. A small change will only give a small change in the stock. For stock levels close to the maximum of the surplus growth function, the long-term loss is so small that it can be justified to move a little to the left of the maximum of surplus growth, until the two terms, the immediate gain and long-term loss become equal.

Immediate gain and long-term loss equal:

THIS: ∆S = ∆G / r

The intuitive explanation is that a fish in the sea represents an (future) investment. If the fish were caught and turned into money, that money would grow at the rate r in the bank.
It is hence a question of investing in natural capital (fish stock growth) versus investment in financial capital (turn fish into money).

Question 18

Q: How can we find the sustainable yield as a function of effort?

Answer 18

1. We have the formula for sustainable stock level: S=K(1-Eq/a).
2. We have the relationship (between…): Y=qES
3. This gives ud the sustainable yield as a function of effort:

Y_sus = aE-βE^2 (a=alpha)
a = qK (a=alpha)
Β=q^2K/a (a=a)

Question 19

Q: Assumptions for Y=qES

Answer 19

1. Fish are uniformly distributed over a given area
2. and more, think

Question 20

Q: Explain the interaction between fishing, the fish stock and stock growth

Answer 20

A popular assumption is that a unit of fishing effort always removes a certain fraction of the fish stock (built on a pretty strong but popular assumption).

1. Y = qES (4.3), where Y is catch of fish, E is fishing effort, S is size of fish stock, q is a coefficient related to the units in how we measure effort and stock, usually referred to as catchability ar availability coefficient.
2. Figure 4.2:
   * G,Y on y-axis
   * S on x-axis
   G(S)=aS(1-S/K) where Y=EqS

• Can be showed with four different effort levels, where the steepest line shows the highest effort
  3. Figure 2:
• Sustainable yield on y-axis
• Effort on x-axis
  aS(1-K/S)=EqS
  Sustainable stock level as a function of fishing effort: S=K(1-Eq/a)
  Y=EqK(1-Eq/a)
• Can again be showed with four different effort levels, which goes vertically up from the x-axis and hits the sustainable curve.

Question 21

Q: Explain the the interaction between fishing, fish stock and stock growth. (1. Biological interaction)

Answer 21

1. Biological interaction
   a) Mathematically:
   Growth of a fish stock: G = aS(1-S/K), where where G is the surplus growth, S is the stock level, a is the intrinsic growth rate, and K is the carrying capacity of the environment (the maximum stock that can be sustained).

b) Graph: [Figure 4.1]
This relationship can be visualized as an inverted U-shaped curve on a graph where the x-axis represents the stock (S) and the y-axis represents the growth (G). This is known as the Surplus Growth Curve of a Fish Stock. The peak of the curve (S = K/2, G = aK/4) represents the maximum sustainable yield (MSY), i.e., the maximum catch that can be continuously taken from a stock under existing environmental conditions.

Question 22

Q: Explain the the interaction between fishing, fish stock and stock growth. (2. Fishing effort and Stock Size)

Answer 22

a) Mathematically:
The catch of fish (Y) is a function of both the fishing effort (E) and the size of the fish stock (S). This relationship is expressed by the equation Y = qES, where q is the catchability coefficient.

b) Graph: [Figure 4.2, left]
On a graph, each level of effort is represented by a straight line that intersects the surplus growth curve at a point that indicates the sustainable yield for that level of effort. The line is steeper for higher levels of effort, and where it intersects with the surplus growth curve determines the sustainable stock level (S_sust = K(1 - Eq/a)) for that level of effort.

Question 23

Q: Explain the interaction between fishing, fish stock and stock growth.

Answer 23

3. Economic Interaction
   a) Graph: [Figure 4.3]
   On a graph, each level of effort is represented by a straight line that intersects the surplus growth curve at a point that indicates the sustainable yield for that level of effort. The line is steeper for higher levels of effort, and where it intersects with the surplus growth curve determines the sustainable stock level (S_sust = K(1 - Eq/a)) for that level of effort.

The economically optimal level of fishing effort is given by the point where the marginal productivity line intersects with the cost per unit of effort line. This is less than the effort resulting from open access and not more than the effort that produces the MSY. This results in the maximization of the difference between revenues and costs, known as the resource rent or fishing rent.

Question 24

Q: Explain the interaction between fishing, fish stock and stock growth. (Summary)

Answer 24

In summary, the interaction between fishing, fish stock, and stock growth is a delicate balance of biological and economic factors. Overfishing can deplete fish stocks and decrease yield, while underfishing can result in missed opportunities for economic gain. Therefore, it is crucial to understand these relationships and manage fishing efforts effectively to ensure the sustainability of fish stocks and the economic viability of the fishing industry.

Question 25

Explain the tradeoff. Show where changes is made by reducing the stock

Answer 25

 Reduce stock by ∆S
 Catch (Y) increases by ∆Y = - ∆S
 And catch value by p(-∆S)
 In the next period, the sustainable catch will decline by G’∆S
 First period gain and long term loss

Question 26

Show by example how we can find the optimal stock level

Answer 26

1. Immediate gain by reducing stock ∆S is p = ∆S = 0.05
2. Value of decline in sustainable yield is p∆G = 0.05*(0.125-0-12375)=0.0000625
   * And occurs in period 1 and all later periods.
3. We then calculate this over an infinite time horizon:
   p∆G/r = 0.0000625/0.05 = 0.00125
   * This sum is negative, but it is infinite
4. It can be justified to go a little to the left of the maximum sustainable yield of the surplus growth curve uniting the two terms, immediatie gain and long term loss becomes equal
   ∆S = ∆G/r
   * For a smaller change in the stock ∆S, the change in sustainable yield ∆G by moving slightly away from the optimal stock level will be approximately equal to the slope of the tangent to the curve at the point (the derivative of that curve), which we denote as G’(S), times the change in the stock which gives
5. G’(S^0) = r
   * Since r >0, the optimal stock level is to the left of the level that produces the maximum yield
   - Fish in the seaproduces G’(S^o) growth
   - But caught fish grow at r rate in the bank
6. G’(S) = a (1-2s)
   0.5(1-20.45) = 0.05
   * So if r = 0.05, the optimal stock is 0.45

Question 27

Difference between sustainable stock level and sustainable yield?

Answer 27

The sustainable stock level (S_sus) represents the size of the fish population that can be maintained over time given a certain level of fishing effort (E). It takes into account factors like the natural growth rate of the fish population and the capacity of the environment (K), as well as the effect of fishing (Eq). In the equation S_sus=K(1-Eq/a), you’re essentially finding the balance point where the growth of the fish population equals the amount of fish being removed by fishing.

The sustainable yield (Y_sus), on the other hand, is the amount of fish that can be caught sustainably, without depleting the fish population over the long term. Here, you’re interested in the actual catch, not just the size of the fish population. That’s why in the equation Y=qE * K(1-Eq/a), you multiply the sustainable stock level by the catchability coefficient (q) and the fishing effort (E). This gives you the sustainable yield as a function of fishing effort.

In other words, S_sus is about the fish that are left in the sea (the sustainable fish population size), while Y_sus is about the fish that are taken out of the sea (the sustainable catch). You need both concepts to manage a fishery sustainably: you need to know how big the fish population should be (S_sus) to support a certain level of sustainable catch (Y_sus).

Question 28

Explain the denotations in S_sus and Y_sus

Answer 28

S: This represents the stock of fish, or the number of fish in the population.

a: This is the intrinsic growth rate of the fish population. This represents the maximum rate of growth when there is no limitation by the carrying capacity.

K: This is the carrying capacity. It represents the maximum number of fish the environment can sustain indefinitely.

Y: This is the yield, or the number of fish caught.

q: This is the catchability coefficient. It represents the efficiency with which the fish are caught and is a measure of how the fishing effort (E) translates into the catch (Y).

E: This represents the fishing effort, such as the number of boats, amount of time spent fishing, or the intensity of fishing methods used.

α (alpha): This represents the maximum sustainable yield, which is the maximum catch that can be removed from the population over an indefinite period of time without causing the population to decline.

β (beta): This represents the change in the sustainable yield as a function of the fishing effort.

Question 29

Explain the logistic surplus growth function G(S)=aS*(1-S/K) and its parameters

Answer 29

The function models how the growth of the fish population changes in response to the size of the population and the carrying capacity of the environment.

a) When the population size is small relative to the carrying capacity, the growth rate is high
b) But as the population size approaches the carrying capacity, the growth rate slows down, eventually reaching zero when the population size equals the carrying capacity

1. G(S): Represents the surplus growth of the fish stock- The amount by which the fish population grows in a given period
2. S: Size of the fish stock at a given time
3. A: intrinsic growth rate for fish popuøation. Represents how quickly the population would grow in the absence of any limiting factors, such as predation.
4. K: The carrying capacity of the environment. The maximum number of fish that the environment can support over the long term. Is determined by factors like availability of food and habitat.

Question 30

Q: Assume that the price of fish and the cost per unit of fish are both constant. What would be the optimal stock size (i) without and (ii) with time discounting? Explain why these two solutions are different. Illustrate in your graph:

Answer 30

1. Graph: G(S) on y-axis, S on x-axis.

a) Without time discounting: S_msy

b) With time discounting:
* Since r > 0, the optimal stock level (S^o) is to the left of the level producing the maximum yield
* Can be justified to go a little to the left until immediate gain and long-term loss is equal ∆S = ∆G/r

1. Immediate gain by reducing stock ∆S is p = ∆S
2. Value of decline in sustainable yield p∆G
3. Calculate over infinite time horizon: p∆G/r
   * The infinite loss. A tradeoff
4. Can be justified to go a little to the left until immediate gain and long-term loss is equal ∆S = ∆G/r
5. G’(S^0) = r
   * Since r >0, the optimal stock level is to the left of the level that produces the maximum yield
   * Fish in the seaproduces G’(S^o) growth
   * But caught fish grow at r rate in the bank
6. G’(S) = a (1-2s)
   * 0.5(1-20.45) = 0.05
   * So if r = 0.05, the optimal stock is 0.45