代写ECON131 Quantitative Methods in Economics, Business and Finance

ECON131 Quantitative Methods in

Economics, Business and Finance

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3

Ecological Footprint (EF)

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 it consumes, absorb its waste and provide infrastructure (Stiglitz et al., 2009, p. 244,

http://www.insee.fr/fr/publications‐et‐services/default.asp?page=dossiers_web/stiglitz/documents‐

commission.htm). Biocapacity is a measure showing the capacity of biosphere to regenerate and provide for

life. More detailed definitions are given on page 5.

Figure 1: Humanity’s ecological footprint by component 1961‐2012

Figure 2: Ecological footprint and Biocapacity per capita 1961‐2012

4

Figure 3: China: Ecological footprint and Biocapacity per capita 1961‐2012

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 Chinese population living at, beyond or below China’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 China? 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 3.4 and for 1961 was

approximately 1).

4. If you assume that biocapacity grows at a constant yearly rate, what is the approximate slope of the

biocapacity relationship with time for China? 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

0.9 and for 1961 was approximately 1).

5. If this trend continues, what will EF and biocapacity be in 2050? (Hint: use your equation from

代写ECON131 Quantitative Methods in Economics, Business and Finance

Definitions: Source: Global Footprint Network

Footprint and Biocapacity Accounting

Footprint and biocapacity accounting helps us answer the basic research question: How much do people

demand from biologically productive surfaces (Ecological Footprint) compared to how

much can the planet (or a region's productive surface) regenerate on those surfaces

(biocapacity)?

What is the Ecological Footprint? What is it composed of?

The Ecological Footprint is the area of land and water it takes for a human population to generate the

renewable resources it consumes and to absorb the corresponding waste it generates,

using prevailing technology. In other words, it measures the "quantity of nature" that we

use and compares it with how much "nature" we have.

The components of the Footprint include:

Cropland: Cropland is the most bioproductive of all the land‐use types and consists of areas used to produce

food and fibre for human consumption, feed for livestock, oil crops, and rubber. The cropland Footprint

includes crop products allocated to livestock and aquaculture feed mixes, and those used for fibres and

materials. Due to lack of globally consistent data sets, current cropland Footprint calculations do not yet take

into account the extent to which farming techniques or unsustainable agricultural practices may cause long‐

term degradation of soil.

Forest land: Forest land provides for two competing services: the forest product Footprint, which is calculated

based on the amount of lumber, pulp, timber products, and fuel wood consumed by a population on a yearly

basis; and the carbon Footprint, which represents the carbon dioxide emissions from burning fossil fuels in

addition to the embodied carbon in imported goods. The carbon Footprint component is represented by the

area of forest land required to sequester these carbon emissions. Currently, the carbon Footprint is the

largest portion of humanity’s Footprint.

Fishing grounds: The fishing grounds Footprint is calculated based on estimates of the maximum sustainable

catch for a variety of fish species. These sustainable catch estimates are converted into an equivalent mass of

primary production based on the various species’ trophic levels. This estimate of maximum harvestable

primary production is then divided amongst the continental shelf areas of the world. Fish caught and used in

aquaculture feed mixes are included.

Grazing land: Grazing land is used to raise livestock for meat, dairy, hide, and wool products. The grazing land

Footprint is calculated by comparing the amount of livestock feed available in a country with the amount of

feed required for all livestock in that year, with the remainder of feed demand assumed to come from grazing

land.

Built‐up land: The built‐up land Footprint is calculated based on the area of land covered by human

infrastructure: transportation, housing, and industrial structures. Built‐up land may occupy what would

previously have been cropland.

What is Biocapacity?

Biocapacity serves as a lens, showing the capacity of biosphere to regenerate and provide for life. It allows

researchers to add up the competing human demands, which include natural resources, waste absorption,

water renewal, and productive areas dedicated to urban uses. As an aggregate, biocapacity allows us to

determine how large the material metabolism of human economies is compared to what nature can renew.

What is a global hectare (gha)?

A global hectare is a biologically productive hectare with world average productivity. Because each unit of

space harbours a different portion of the global regenerative capacity, each unit is counted proportional to its

global biocapacity share. For this reason, hectares are adjusted proportionally to their productivity and are

expressed in global hectares

6

Environmental Resource Management

Motivation

The Atlantic cod is a massive fish that can grow up to 2m in length, weigh up to 96 kg, and can live up to 25

years. Before WWII, there were more than a million tonnes of Atlantic cod living in the Arctic. Due to

overfishing, by 1990 this number declined to 118,000, on the brink of collapse.

Humans depend on biotic resources like fish, animals and forests. Ensuring these resources are not over‐

exploited is absolutely vital, not just for the health of the planet, but for our own survival. Finding the right

level at which to exploit these resources is important: on one hand, if harvesting quotas are set too low,

millions of people might go hungry. But if resources are over‐exploited, there is the risk that the population

may collapse entirely, with disastrous consequences for human livelihoods.

In this assignment, we analyse three models of biological populations, and find methods for working out how

to exploit them sustainably.

Part A

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 b of a population will reproduce,

and some fixed proportion d will die.

The model describes the evolution of a population over time, t.

ܰሺݐሻ ൌܰ݁ሺିௗሻ௧

, (1)

where N0 is the population at time t=0.

1. If the initial population N0=2000, the annual birth rate b=0.03 (3% exponential birth rate) and annual

death rate d=0.02 (2% exponential death rate), what is the population after 1 year? Explain this

result.

2. Find an expression for the growth rate of the population in terms of b, d and N(t).

Hint: remember that 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?

代写ECON131 Quantitative Methods in Economics, Business and Finance

Part B

Planet Issues – Unlimited population growth with

harvesting

The model in equation (1) describes a population untouched by humans: there is no harvesting. We can

model what happens if we harvest H individuals from the population each period. The unlimited population

growth with harvesting model is given by

ܰሺݐሻ ൌܽ݁ሺିௗሻ௧

ுିுሺ್షሻ

ሺିௗሻ

, (2)

where b is the birth rate, d is the death rate, and H is the number of individuals taken from the population in

each period.

1. Confirm that the rate of population change is

ௗேሺ௧ሻ

ௗ௧

ൌሺܾെ݀ሻܰሺݐሻെܪ. (3)

Hint: differentiate (2), then add (+H‐H), and rearrange so that you can substitute (2) back in.

(5 marks)

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

3. Let N0=N(0)=2000, b=0.05 and d=0.02. At what rate can the population be harvested sustainably?

Hint: The sustainable 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 4% 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?)

8

Part C

Planet Issues – The Verhulst model of ecological growth

We now consider a slightly more sophisticated version of the above model. Leaving harvesting to one side for

a minute, we will consider what happens when there is a limit to growth. The previous model assumed that

populations could continue to grow indefinitely. Empirical research suggests that there tends to be a limit to

the size of population that an environment can support; we call this limit the carrying capacity, K.

A model linking these ideas was first proposed by a French mathematician Pierre Verhulst in 1838. The

Verhulst model can be written as:

(4)

where N0 is the initial population level (Bacaër, 2011). For simplicity, we’ll write the net reproduction rate

simply as r, rather than in terms of births and deaths.

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 t?

How does the population level change when the growth rate r increases?

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.

(5 marks)

4. Does the population grow faster when it is above the environment’s carrying capacity (N>K), or below

it (N<K)?

5. A pond has a carrying capacity of 200,000 fish. Its population this year is 190,000 individuals. If the

reproduction ratio is r = 1%, 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 95,000. 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.

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代写ECON131 Quantitative Methods in Economics, Business and Finance