# Quantitative Methods in Economics, Business and Finance Part A: Ecological Footprint The Ecological.

Quantitative Methods in Economics, Business and Finance

Part A: Ecological Footprint

The Ecological footprint (EF) measures how much of the regenerative capacity of the biosphere is used up by human activities. It is the sum of productive land and water area required to support the population and provide the resources the population consumes, absorb its waste and provide infrastructure.

Biocapacity is a measure showing the capacity of biosphere to regenerate and provide for life. More detailed definitions are given on page 5.

1. According to the EF, is the human population living at, beyond or below the Earth's natural biocapacity? For how long has this been the case? Is this sustainable?

2. According to the EF, is the Australian population living at, beyond or below Australia's natural biocapacity? For how long has this been the case? Is this sustainable?

3. If you assume that EF grows at a constant yearly rate, what is the approximate slope of the EF relationship with time for Australia? And what is its units? Give the equation of, and sketch this line, with EF on the vertical axis and year on the horizontal axis. (EF for 2012 is 9.3 and for 1961 was approximately 8).

4. If you assume that biocapacity grows at a constant yearly rate, what is the approximate slope of the biocapacity relationship with time for Australia? And what is its units? Give the equation of, and sketch this line, with biocapacity on the vertical axis and year on the horizontal axis. (biocapacity for 2012 is 16.6 and for 1961 was approximately 29).

5. If this trend continues, what will EF and biocapacity be in 2050? (Hint: use your equation from questions 3 and 4.) What will biocapacity minus EF be in 2050? Will there be a deficit or reserve?

6. Use the data presented in Figure 1 to suggest a policy to make land use sustainable.

Part B: Planet Issues – Unlimited Population Growth

One way to think about the growth of some biomass – whether a petri dish full of bacteria or a river full of fish – as some fixed proportion of the existing population. The unlimited population growth model is a simple model which assumes that, in each period, some fixed proportion of a population will reproduce, and some The model describes the evolution of a population over time t.

1. If the initial population is N 0 = 1000, the annual birth rate b = 0.02 (2% exponential birth rate) and annual death rate d = 0.01 (1% exponential death rate), What is the population after 1 year? Explain the result.

2. Find an expression for the growth rate (population change) of the population in terms of time and the parameters. Hint: differentiating with respect to time gives a rate of change.

3. If d < b,="" will="" the="" expression="" you="" found="" in="" the="" previous="" part="" be="" positive="" or="" negative?="" how="" would="" you="" interpret="" this="" fact?="">

Part C: Planet Issues – Unlimited Population Growth with Harvesting

1. Confirm that the rate of population change is

dN(t)/dt = (b-d)N(t) – H.

2. Give the meaning of the expression in equation (3).

3. Let N 0 = N(0) = 1000, b = 0.05 and d = 0.01. At what rate can the population be harvested sustainably? Hint: The sustain harvest, H, will be the value that causes no change in the overall population, i.e. dN(t)/dt = 0.

4. Suppose an ecosystem is being sustainably harvested at exactly its replacement rate, and the population is constant. Now suppose the population experiences a brief disease epidemic, which causes 5% of individuals to perish. What will happen to the population if harvesting continues at the same rate?

5. Suppose a population of fish that follows this model is being harvested sustainably, as before. What will happen in the long run if, one day, a fisherman decides to throw one of the fish back?

6. This type of model is often said to have a “knife-edge” equilibrium. Explain what this means.

7. Governments often levy fishing quotas in areas where populations are at risk. Does this model shed any light on the effectiveness (or otherwise) of these policies? (Hint: what if someone cheats?)

Part D: Planet Issues – The Verhulst model of ecological growth

1. What is the growth rate if the initial population is zero?

2. When the population is initially at its carrying capacity, i.e. N0 = k, what is the population at time ? How does the population level change when the growth rate r increases?

3. Differentiate (4) to show that the rate of population growth at time t is

dN(t)/dt = rN(t)(1 – N(t)/K)

Remember to use the quotient rule to differentiate (4). You will find it easier if you carefully look for the terms that you expect to see in the final expression, and factor those out.

4. Does the population grow faster when it is above the environment's carrying capacity (N < k),="" or="" below="" it="">

5. A pond has a carrying capacity of 100,000 fish. Its population this year is 90,000 individuals. If the reproduction ratio r = 2%, and the population follows the Verhulst population model, how many fish can be harvested per year, leaving a constant population?

6. Suppose that, due to unlicensed fishing, the population falls by half, to 45,000. How many fish can be harvested per year, in order to maintain a stable population? Explain the difference between the previous question and this question.

Attachment:- assignment.rar