alllllt Flashcards
Explain what is meant by biological versus economic overexploitation of fish stocks (text)
Biological:
* Fishing activities reduces the fish population.
Economical:
* Fishing activities exceed the sustainable limits of the fish population, driven by economic motivations
Explain what is meant by biological versus economic overexploitation of fish stocks (Draw graph, model and explain S and S_msy).
Graph: Y-axis is G(S). X-axis is S.
* Fully exploited at S_smy
* Overexploited to the left
* Underexploited to the right
Model: G(S)=aS(1-S/K).
Biological overexploitation:
If S < S_msy
Economically overexploitation
Can be even if S > S_msy.
Depends on:
a) how costs per unit are related to the fish stock
b) how strong future revenues are discounted.
Don’t have to be economically overexploited even if S > S_msy
Explain the different variables in the growth function
G(S)=aS(1-S/K)
a – intrinsic growth rate
S – current size of fish stock
K – carrying capacity of the environment
Could biological overexploitation be economically justified? (Mathemathics)
We reduce the stock to a slightly lower value
(1) S - ∆S
This results in an immediate increase in fish catch
(2) ∆S
The smaller stock is less productive, leading to a decline in the surplus growth function
(3) G(S - ∆S)
We can then find the change in growth
(4) ∆G = G(S) – G(S - ∆S)
The change in growth ∆G is the loss in the growth for every subsequent period. If we assumes this to perpetuity, we can calculate the PV as
(5) ∆G/r
This creates a tradeoff: immediate gain in fish catch versus long term surplus growth. We find the optimal level when we put these equal to each other
(6) ∆S = ∆G/r
The change in sustainable yield (∆G), by moving slightly from the optimal stock level, will be approximately equal to the slope of the tsngent to the surplus growth curve times the change in the stock ∆S
(7) G’(S^o)=r
Since r > 0, the optimal level is left of the level that produces maximum sustainable yield.
Find out what the optimal level is to harvest when r>0 (Mathematical)
(1) G(S) = aS(1-S/K)
(2) G’(S) = a(1-2S)
This function describes the slope of the surplus growth function at any given point. We then set this function equal to r to find the stock level to find the stock level that produces the stock level where the biological growth equals the economic growth rate
(3) G’(S^o)=r
(4) a(1-2S)=r
Give an example to show why S_smy might not be the optimal
Surpluss growth function: G(S)=aS(1-S/K). Also, lets say that a 0.5, K=1, S_msy= 0.5
G(0.5) = 0.5*0.5 * (1 - 0.5/1) = 0.125
We now reduce the stock from 0.5 to 0.45.
G(0.45) = 0.5 * 0.45 * 0.55 = 0.12375
Annual loss is (a)-(b) 0.125 – 0.12375 = 0.00125
The present value of the loss in perpetuity is 0.00125/r. With 5% discount rate, we get 0.025.
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
Find sustainable stock level and sustainable yield as a function of effort. Then draw 2 graphs that explain 2 different levels of effort
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
Graphs at p. 70
Draw Graphs “high” versus “low” cost. Give an explanation.
p. 72
Graph 1:
* Sustainable yield initially increases with effort. Reaches a peak before it declines. This reflects the biological limits of the fish stock
* Assumes linear costs
* Low cost results in more effort than high cost. With low cost, there is big competition and much effort
* E1^0 and E2^0 are optimal points of effort. Difference between revenue and cost are maximized
* E1* and E2* is when they fish until profits = costs
Graph 2:
Marginal Productivity and Average Productivity
* AP – Total output / Total effort
* MP – The additional output of one more effort
FISH LEFT IN SEA vs FISH CAUGHT AND SOLD
FISH LEFT IN THE SEA:
* Contribute to the growth of the population at rate G’(S^o)
* Can be considered as “return on investment” for leaving in the sea
FISH CAUGHT AND SOLD
* When they are caught and sold, the money can be put in the bank where it will grow at interest rate r
GRAPH:
p. 6 graph skrivebok
What is the different goals with and without r
- Without r – Goal is to maximize biological growth
- With r – goal is to maximize economic value
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
Assumptions for the Harvest function
- 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) - 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) - 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
Explain what is meant by Open-access fisheries
- Anyone who wants to fish can do so.
- No restriction or regulation on the amount of fish than can be caught, or the number of people that can participate
- Given that fish stocks are limited, open-access fisheries are often subject to a phenomenon known as the “tragedy of the commons,” where the resource is overexploited until it becomes economically unviable or ecologically exhausted.
- “Tragedy of the commons”
Draw graphs that show the impact of subsidies
Question 4 i skrivebok
R=S_(t-1) + G (S_(t-1). A =0.5 and K=1. What is the stock that maximizes sustainable yield? What is the maximum sustainable yield?
easy
R=S_(t-1) + G (S_(t-1). A =0.5 and K=1. What is the optimal stock to be left after fishing at 5 percent rate of discount? Why is this different from the stock that maximizes sustainable yield? Explain carefully. What is the sustainable yield of this stock?
Easy
R=S_(t-1) + G (S_(t-1). A =0.5 and K=1. Beta = 0.75.
1) How much of the stock would this country leave behind after fishing if β = 0.75?
2) How much would the other country leave behind?
Classic scenario of game theory. Two countries have to decide how to maximize their individual payoffs based on their expectations about what the other entity will do.
Country 1:
* Beta = 0.75
* Aims to maximize its benefits from the fishery, given its initial stock (R). The fishery’s value is determined by two components:
a) The immediate benefit, which is the profit from fishing
b) The future benefit (growth of the stock)
Maximum Condition:
1 + r = B * (1 - dG/dS)
Solving the derivate, gives:
dG/dS = ((1 + r) / B) - 1
Country 2:
Strategy:
* The share of the stock is smaller, so they have a smaller incentive to leave fish behind for future growth!!!!!
* In fact, given that the stock grows and breeds as a unit, Country 2 has an incentive to free ride on Country 1’s conservation efforts!!!!
Maximum Condition:
1 + r = (1 - B) * (1 - dG/dS)
Solving Derivative gives
dG/dS = ((1 + r) / (1 - B)) – 1
SOLVE FOR COUNTRY 1:
0.5*(1-2S)=(1+r)/B – 1
S= 0.1
Country 1 leaves 0.1 left
Q: Compare Country 1 and 2’ different conditions in the game theory example.
- compare formula - inherently incompatible
* left side the dG/dS represent growth of fish stock- this is the same
* Right side different. Smaller stock gives bigger for C2 - C2 prefer small stock than C1. But same stock, so both cannot get what they prefer
- only solution is C2 leaves nothing behind after fishing, thus becoming a free rider.
* they gain very very very little from leaving fish in the ocean
R=S_(t-1) + G (S_(t-1). A =0.5 and K=1. Beta = 0.75, S1=0.1. What would be the catches of the two countries in this situation? Explain the outcome carefully.
Country 1’s Catch
Given that Country 1 will leave a stock S1 = 0.1 behind, the returning stock each period will be:
R = S1 + G(S1) = 0.1 + 0.5 * 0.1 * (1 - 0.1) = 0.145
Country 1’s share of this returning stock is 0.75, so the amount of fish that Country 1 takes, or its “catch”, is its share of the returning stock minus the amount it leaves behind:
Catch_1 = 0.75 * R - S1 = 0.75 * 0.145 - 0.1 = 0.00875
Country 2’s Catch
Country 2, being a free rider, leaves nothing behind after fishing. So its catch is simply its share of the returning stock:
Catch_2 = 0.25 * R = 0.25 * 0.145 = 0.03625
Note that Country 2’s catch is larger than Country 1’s catch, even though its share of the stock is smaller. This is because Country 2 is taking advantage of the conservation efforts of Country 1, essentially getting a “free lunch”.
R=S_(t-1) + G (S_(t-1). A =0.5 and K=1. Beta = 0.75. Would the minor country accept a share of the catch in the cooperative solution (cf. b, 0.12375) equal to its share of the stock (1 – β)?
The cooperative solution mentioned in Question b suggested that the stock should be fished down to a size that maximizes the present value of the fishery, which led to a sustainable yield of 0.12375.
In this cooperative scenario, let’s see if Country 2 would accept a share of the catch equal to its share of the stock (1 – β).
Cooperative Catch for Country 2
In a cooperative situation, the sustainable yield would be shared proportionately between the two countries based on their shares of the stock. Country 2, owning 25% of the stock, would therefore be offered 25% of the sustainable yield:
Offered Catch_2_coop = 0.25 * 0.12375 = 0.030938
Comparison of Cooperative Catch vs Non-Cooperative Catch for Country 2
Now, let’s compare this offer to what Country 2 was able to catch on its own in the non-cooperative scenario (Question d).
Recall that in the non-cooperative scenario, Country 2 caught 0.03625 of the stock. In the cooperative scenario, it’s being offered 0.030938.
Clearly, Country 2’s catch under the cooperative scenario is less than what it got in the non-cooperative scenario:
0.030938 (cooperative) < 0.03625 (non-cooperative)
Therefore, Country 2 would not accept this offer. It would prefer to continue the non-cooperative approach because it’s catching more fish that way.
In analyzing the optimal rotation problem for trees, we often use a general growth function f(t) where f(.) is the amount of timer and t is the age of the trees. How would you specify this function if you applied it to a brood of fish?
Must take unique trats of fish into account.
* Growth period is short (1.5 – 3 years)
* Growth is seasonal, could dominate longer term growth
* Fish must be marketed before they put most energy into growing roe and milt
* Prices and demand may be seasonal and affect optimal T more than just growth
See formula PPP
The break of cooperation
1: PV with cooperation
We assume that fishing occurs in periods and that the stock in t, R_t, depends on how much that was left in the previous periods (S_(t-1):
(1) R_t = S_(t-1) + G(S_(t-1))
In the cooperative solution, the countries have agreed to leave behind an economically optimal fish stock, S^o, and fish from that optimal stock G(S^o).
The optimal stock and its sustainably yield will produce a profit per period of π^o(S^o)
The PV of profits for each country will be
(2) V^o = (π^0(1+r)) / (rN)
2: One country abandon
Now, one country abandons. The country fishes down to S* instead of S^o. This is only found out when the period is over.
The best response from the other countries is to do the same.
Stock level S* will produce a profit of π(S)
The PV for each country is
(3) V* = (π(S))/(rN)
3: When is it profitable to break the cooperation?
Breaking the cooperation is profitable if
V* + D > V^o
Where D is the profit the country gets in the period it deviates from the cooperative agreement. This is
D = [π^o(S^o))/N] + T (S^o + S*).
In the first part, the country gets its share of the cooperative profit that is π^o/N. In addition, it benefits T from fishing down the stock from S^o to S. This is solely for the deviating country. So, for it to be profitable:
T (S^o – S) > (π^o(S^o) – π(S))/rN
The one-time gain T must be greater than the PV of the difference between the cooperative an non-cooperative profits from the period after the deviation. This can specially happen if r or N is big enough.
What are subsidies in the world of fisheries?
Financial aid provided by governments to support industries. Can be new boats, artificially raise the price of fish etc.
What has the effect of subsidies been?
Have a bad effect. Specially in Open-Access
1. Lower the costs which encourage more fishing
2. Goes from old equilibrium E* to new that is E*_s, which implies more fishing effort
3. Increased effort can seem good in short term but is detrimental in long term as it leads to overfishing. The underlying problem – poverty in the fishery – isn’t solved but rather perpetuated.
Effects of Price Subsidies and Example
They rise the price of fish.
1. Sustainable yield curve get displaced upwards and promting and expansion of fishing efforts. Similar to direct subsidies, it can lead to overfishing
A classic example is Norway, where price subsidies were provided from the 1950s to the 1990s. Yet, these subsidies did little to enhance the well-being of fishermen or the sustainability of the fish stocks.
Q: The Faeroese Example:
- 1970s and 80s. Faroese government provided subsidies to the fishing industry
- Led to overcapacity and decline in fish catches
- Nearly bankrupted the government
Explain the Beverton-Holt model and formula
R = aS / (1 + S / K)
* R represents recruitment, the number of new individuals added to the population.
* S represents the spawning stock biomass, the total weight of sexually mature fish in the population.
* a, K are parameters of the model, usually determined by fitting the model to empirical data.R is recruitment and S is the spawning stock, and a, K and b are the estimated parameters
A model to understand how fish populations change over time. The basic idea is that there’s a relationship between the number of adult fish (the “stock”) and the number of baby fish, or “recruits,” that are added to the population each year. This relationship is influenced by the environment, predation, disease, and other factors.
Q: Ricker Formula
R = aS expo(-bS)
R is recruitment and S is the spawning stock, and a, K and b are the estimated parameters
* “b” is a parameter that represents how much the survival of the baby fish decreases as the adult population increases (the degree of competition).
The key feature of the Beverton Holt model
The key feature of the Beverton-Holt model is that it assumes density-dependent survival, which means that as the number of adult fish increases, each individual offspring has a lower probability of survival. This model results in an asymptotic relationship between spawning stock and recruitment, where increasing the spawning stock beyond a certain point does not result in a substantial increase in recruitment.
Beverton – The Principle of Density Dependence
The model assumes that when the adult population is low, there is less competition for resources, so a higher proportion of the baby fish survive. However, as the adult population gets larger, competition increases and a smaller proportion of the baby fish survive. This is a concept known as “density dependence.”
Q: Beverton – Assumptions
The Beverton-Holt model assumes that the environment is stable and that the rate of survival of the baby fish is solely determined by the number of adult fish.
Q: Draw and explain the shape of the Ricker model
In the Ricker model, recruitment initially increases with spawning stock size but eventually decreases when the spawning stock size gets too large. This is because the model assumes that as the population grows, the resources per individual decrease leading to a decline in survival and hence recruitment. This leads to an inverted-U shaped curve, unlike the Beverton-Holt model.
Q: Sketch the sustainable yield curve of the Beverton-Holt-model and explain its shape
Page 89.
The curve in your Beverton-Holt graph represents how much fish can be sustainably caught (yield in million tonnes) as fishing mortality rate (F) increases.
1. Initial Rapid Increase: At very low levels of F (near 0), an increase in F leads to a rapid increase in yield. This is because as more fishing effort is expended (higher F), more fish are being caught.
2. Early Peak: The curve reaches its peak at a relatively low level of F (~0.25). This is the point of Maximum Sustainable Yield (MSY), where the fishing mortality rate allows for the highest yield that can be caught without depleting the fish population over time.
3. Slow Decline: After the peak, yield decreases as F increases. This indicates overfishing – fishing effort continues to increase but the yield declines because the fish population cannot sustain such high mortality rates.
Q: Draw and explain the Ricker graph
Page 89
The curve on your Ricker graph follows an inverted U-shape, reflecting how yield changes as fishing mortality rate increases.
Key Points of the Ricker graph:
1. Initial Increase: Similar to the Beverton-Holt model, the yield increases rapidly with an increase in F at the beginning. The increased fishing effort initially results in more catch.
2. Peak and Rapid Decline: The yield reaches a peak (at MSY) and then falls sharply. This sharper decline as compared to the Beverton-Holt model signifies a more dramatic effect of overfishing. It shows that, beyond a certain point, the increase in fishing effort significantly depletes the fish population, leading to a rapid decline in yield.
Q: Difference between Ricker and Beverton
- Density-dependence
* The Beverton-Holt Model is a density-independent model (the growth of the population does not depend on the population’s initial size)
* Ricker Model is a density-dependent model. (the population growth rate changes as the population size increases, typically slowing down due to resource limitation) - Response to overfishing
* Beverton-Holt: Gradual decline after MSY. This suggests that even if fishing pressure continues to increase, the decrease in population size and yield is slow.
* Ricker: displays a sharp, rapid decline in yield past the MSY point. This implies a more drastic response to overfishing – a small increase in fishing pressure could cause a large drop in population size and yield.
Both graphs and models help understand the dynamics of fish populations under fishing pressure, but they imply different levels of sensitivity to overfishing
