Understanding Ecological Footprint And Biocapacity

1. Human have been living beyond their biocapacity since approximately 1970. This is unsustainable since the production over capacity implier borrowing resources from the future.
2. The US population been living beyond their biocapacity since approximately 1970. This is unsustainable since the production over capacity implies borrowing resources from the future.
3. The Ecological Footprint is 0.1 for every 40 years  i.e 0.03% per capita per year. (The assumption here is that the yearly rate is constant , not compounded)
4. The growth rate of the biocapacity is -0.76% i.e the biocapacity is diminishing at the the rate of 0.76% per year.
5. The Ecological footprint will be at 1.35 per capita and the Biocapacity will be at 5.187 Per capita per annum

1. There should be progressive taxation for built up land i.e governments could tax a propert based on how much built up land is being used and place them in brackets along the line of income brackets. Just as income brackets are subject to progressive taxation, so should be built up land.

1. The population after a year N₁= 1505012525. The population grows at the rate of slightly over 1% per annum.
2. The growth rate of any population is the first derivative of the total population. Thus, keeping N₀ as the constant, applying derivation w,r,t to “t”
   1. The rate of growth is {d N(t) / d(t)} = (b-d) e^(b-d)t
3. If b is less than d, then there will be a negative effect. This implies that the population is declining since the death rate is higher than the birth rate.

1. The rate of population change is the first derivative of the population growth rate.
   1. Therefore, differentiating w.r.t “t”

{d N(t) / d(t)} = (b-d) e^(b-d)t  –  H

1. The population grows at a rate that is the product of the current population survival rate (birth Rate – Death Rate) and the rate of the evolution of the population less the population that has been harvested. Population that is harvested will not have a growth rate while the population that survival growth rate of the population will continue to grow at the rate of evolution. In very simple terms, out of the existent population, the population that is not harvested is the remainder population. Out of this remainder population, the bet population is the difference between the birth rate and the death rate. In every compounding period, this fraction of the total population will grow at the rate of evolution. Thus, this is the rate of growth.
2. Sustainable level of harvesting is when the growth of population of fish is not affected i,e when,  {d N(t) / d(t)} = 0
   1. I,e (b-d)N (t)  –  H = 0  -à equation 3
   2. Therefore, where ,
   3. (b-d)N (t)  = H
   4. (0.06 – 0.02) 3000 = H
   5. e. 0.04 X 3000 = H
   6. I,e H= 1200. The sustainable level of harvesting is 1200.
3. At the optimal rate of harvesting,
   1. (b-d)N (t)  = H
   2. Let b-d = x
   3. Therefore, the new population will be
   4. (x-0.06) N(t) = H
   5. Therefore,
   6. H= N(t)x – 0.06N(t)
   7. The Harvesting does not change. Hence, the new population is at N(t)x – 0.06N(t)
4. The results of the fisherman throwing the fish back will depend on whether the fish population is over the tipping point of extinction. If the population of fish is over the tipping point (over the knife edge), then the effect of the fisherman’s actions will result in a simple addition to the population of the fish. However, if the Fish is below the tipping point , then the fish population could go from the tipping point of under populated to over populated.
5. A Knife Edge Equilibrium implies an equilibrium that is unstable. If everything is not at an exact 50-50 % balance , then the entire equilibrium can tip to one side. In such an equilibrium there is no plateau. Market entities are all on one side of the edge or another. Once the tipping point is achieved , any growth further from that will lead to all market entities sliding over to the other side. This implies, that if the perfectly stable point is not achieved, everything may go wrong.(Ellison & Fudenberg, 2012)
6. Fishing quotas will control the rate of harvesting. According to that model, if the fishing quota is less than or equal to the sustainable harvesting level, then the fish population is maintained. However, the effectiveness of such a policy will be determined on how many people follow the quotas. Very often, quotas are levied based on restriction n areas of fish and not based on the number of fish that are collected.

1. If the initial population state is zero, then there is no one to reproduce . Hence, the growth rate is zero. (Mathematical proof given in the next question)
2. Let m= 1/k

1. {d N(t) / d(t)} = r. N(t) {1- N(t)}
2. Now, multiplying with m or “1/K”
3. {d N(t) /N (t) (1-mN(t)}= rd(t)
4. Integrating both sides
5. ∫LHS = ∫{d N(t) /N (t) (1-mN(t)}
6. Using Partial Fraction decomposition,
7. {1/ {N(t) (1-m N(t)}= A/ {N(t) +B (1-mN(t)}
8. Using elimination method to obtain a common denominator..
9. 1= A(1-m N(t)+ 1}+BN (t)
10. Let N(t)= 0
11. 1= A (1) +B(0)
12. A= 1
13. Let N(t)=  1/m
14. 1= A(0) +B (1/m)
15. 1= B/m
16. Therefore B= m
17. ∫{1/ N (t)  +m / (1-m N (t)}d N (t) = ∫rdt
18. Ln lN (t)  l – ln (1-m N (t)l = rt +k
19. Ln lN (t)  / m N (t)  } =e^c e^rt+k
20. Let e^c=  C
21. Ln lN (t)  / m N (t)  } = Ce^rt
22. N (t) =  {1- mN (t)  }  Ce^rt
23. N (t) +  mN (t)  } =   Ce^rt
24. N (t) +  mN (t)  } =   Ce^rt
25. Multiplying both sides by K
26. N (t) ={K Ce^rt/  K +Ce^r)

1. According the Verhulst model, the growth rate =  rN(t) (1- N(t)/ K). This is the sustainable amount of fish that can be harvested as harvesting more than this point will cause the fish population to diminish.
2. The environment grows faster when the carrying capacity is greater.
3. Using this formula, 2500 fish can be harvested per year.
4. 5  fish can be harvested per year. In the above model, the number of fish increases, since the reproduction ratio is high. In this model, the net reproduction ratio is low, since there is an increase in the death rate . (r= b-d or the difference between births and death)

Ellison, D., & Fudenberg, D. (2012, February 28). KNIFE-EDGE OR PLATEAU: WHEN DO MARKET MODELS TIP?*. Retrieved from MIT Libraries: https://economics.mit.edu/files/7608

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