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).
In An Essay on the Principle of Population, as It Affects the Future Improvement of Society, With Remarks
on 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. 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 initially a0 (when t=0) and that it grows by a ratio r every year. Write
down an expression for the population in year t. (2 marks)
3. Malthus suggested that food supplies are growing at an arithmetic rate. Assume that the annual
food supply is initially b tonnes per year, and that every year it increases by m tonnes. Write
down an expression for the food supply, in tonnes, in year t. (2 marks)
4 4. Compare the expressions you found in questions (2) and (3). Let the initial population a 0 be
1,000, let the population growth ratio r be 1.05, the initial annual food supply b be 2,400 tonnes,
and the annual increase m be 190. On the same set of axes, where time is the horizontal axis and
people/tonnes are the vertical axis, plot the values for t=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 that r=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? Provide some evidence from your own research to
support your answer. (5 marks)

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