## Economics Assignment – 2017 Section B: Malthusian Disaster

Economics Assignment – 2017

Section B: Malthusian Disaster

In 1793 the political economist Thomas Malthus noticed that that population growth in the United States had been doubling every 25 years (which is geometric growth),
but that the level of food production had only increased by a fixed amount each year (which is arithmetic, or linear growth).
InAn Essay on the Principle of Population, as It Affects the Future Improvement of Society, With Remarkson the Speculations of Mr Godwin, Mr Condorcet and Other
Writers, he wrote:
[. . . ] the power of population is indefinitely greater than the power in the earth to produce subsistence for man. Population, when unchecked, increases in a
geometrical ratio. Subsistence increases only in an arithmetical ratio. A slight acquaintance with numbers will shew the immensity of the first power in comparison of
the second. By that law of our nature which makes food necessary to the life of man, the effects of these two unequal powers must be kept equal. This implies a strong
and constantly operating check on population from the difficulty of subsistence. This difficulty must fall somewhere; and must necessarily be severely felt by a large
portion of mankind.

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World population growth 10,000BC-2,000AD.

Source: US Population Bureau/Wikimedia Commons

1. Look at the graph of world population growth above. Does the growth in the population look arithmetic (linear), or geometric? (1 mark)
2. Assume a population is initiallya0 (when t=0) and that it grows by a ratior every year. Write down an expression for the population in yeart. (2 marks)
3. Malthus suggested that food supplies are growing at an arithmetic rate. Assume that the annual food supply is initiallyb tonnes per year, and that every year it
increases bym tonnes. Write down an expression for the food supply, in tonnes, in yeart. (2 marks)
4. Compare the expressions you found in questions (2) and (3). Let the initial population a0 be 1,000, let the population growth ratior be 1.05, the initial annual
food supplyb be 2,400 tonnes, and the annual increasem be 190. On the same set of axes, where time is the horizontal axis and people/tonnes are the vertical axis, plot
the values fort=0,10,20,30,40 and 50 (4 marks)
5. Now, using the parameters in the previous question, suppose that each person consumes one tonne of food per year. During which year will the population begin to
experience food shortages? Derive your answer mathematically, rather than graphically. You may assume there is no food stored from year to year. (4 marks)
6. Find the year that the population’s demand for food exceeds 22,000 tonnes per year. (4 marks)
7. Suppose that the population growth rate is slightly lower, and thatr=1.01. Now find the year that the population’s demand for food exceeds 22,000 tonnes per year.
(3 marks)
8. Practically speaking, is it inevitable that, if food is growing arithmetically (m>0) and population geometrically (r>1), that food supplies will always run out? In
reality, does it look like earth is heading towards a Malthusian Disaster? Providesome evidence from your own research to support your answer. (5 marks)

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Section C: Solar Panels Investment

Suppose a family consumes 15kWh of power per day. Concerned about its carbon footprint, the family would like to ensure 50% of its electricity comes from renewable
sources. Rational and economically minded, the family would like to find the cheapest way to do so, and its planning horizon is the next 15 years. Assume the panels do
not degrade over time, and at the end of 15 years, the solar panels will have zero residual value. Also assume that electricity provided by the solar panels has a per
unit cost of zero (i.e. zero variable cost).

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System Type

Electricity Generated / day

Installation Cost

1 kW system

3.9 kWh / day

\$6 000

1.5 kW system

5.85 kWh

\$7 000

2 kW system

7.8 kWh

\$8 000

3 kW system

11.7 kWh

\$11 000

4 kW system

15.6 kWh

\$14 000

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Typical solar systems available in Sydney. Source: Clean Energy Council
To make this investment, the family would withdraw from its cash management trust, which is expected to return a steady 6.75% per year, compounded annually, for the
foreseeable future. The remainder of the family’s power is provided by the electrical grid. They can buy three different `types’ of electricity from the grid: non-
renewable, 50% renewable or 100% renewable. They face the following prices for energy they purchase from the grid:

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Item

Units

Non-renewable

50% Renewable

100% Renewable

First 1000 kWh

\$ per kWh

\$0.2684

\$0.2838

\$0.2992

>1000 kWh

\$ per kWh

\$0.2805

\$0.2959

\$0.3113

Supply charge

\$ per day

\$0.6908

\$0.6908

\$0.6908
Source: Origin Energy

You may assume that electricity prices remain constant, and inflation can be ignored. Assume the power generated by the solar panels is the same year-round.
1. Which is the cheapest solar system type that would provide 50% of the family electricity consumption? (1 mark)
2. Assuming that the solar system performs as advertised, what is the family’s quarterly bill from its energy provider? Assume there are 91 days in a quarter. (2
marks)
3. Using an annual discount rate of 6.75% and ignoring inflation, what is the present value of 15 years’ worth of electricity bills? (2 marks)
4. Other than buying solar panels, what is the quarterly cost of the next best alternative, which still provides 50% renewable energy? (3 marks)
5. What is the present value of 30 years’ worth of this solution? Again, ignore inflation, and use a discount rate of 6.75%. (3 marks)
6. Conditional on using at least 50% renewable energy, what is the net present value (NPV) of purchasing the solar panel system named in question (1)? Hint: to answer
this question you will need to consider the present value of renewable energy sources and the present value residual electricity bills along with any associated
installation costs. (4 marks)
7. Interpret your result from question (6). Is buying solar panels a good idea? Explain your answer. (2 marks)
8. Under the same assumptions as above, assume the family is not committed to purchasing renewable energy (i.e. the family is happy with consuming non-renewable energy
from the grid.) What is the NPV of purchasing solar panels now? (3 marks)
9. Will the family in question (8) purchase the solar panel system named in question (1)? (2 marks)
10. Some governments have offered subsidies to consumers to install solar panels in their homes. Why? (3 marks)