Malthus Theory: Thomas Malthus (1766 − 1834) was an English political economist who was concerned about what he saw as the decline of living conditions in nineteenth century England.

Question

Malthus published “An essay of the Principle of Population” in 1798. He blamed this decline on the rising of human population and the inability of feed resources. According to Malthus, the population will increase by 3 % every year and the feed resources will increase by 0.48 million every year

Year Population (million) Feed resources (million of people)

1800 10 12

1801 10.3 12.48

1802 10.609 12.96

1. According to Malthus’ theory:

a. Which kind of sequence 𝑃 describe the population growth? Give its characteristics.

b. Deduce the population in 1830 and in 1850 c. When did the population amount to 30 million?

2. According to Malthus’s theory:

a. Which kind of sequence 𝑅 describes the feed resources? Give its characteristics.

b. How many people did the resources feed in 1830 and in 1850

c. When did the resources feed more than 40 million people? Use an algebraic method.

3. Here is a graph extract from Malthus’ essay

Malthus' Basic Theory --> https://www.google.com/search?q=malthus+basic+theory+graph&rlz=1C1CHBF_frFR937FR937&sxsrf=ALeKk01rQwPgOAWX4GXTWXz0LZ4ZAAsglw:1616178988581&tbm=isch&source=iu&ictx=1&fir=AO-3uctMi1z2RM%252CivVYjwIPKf74iM%252C_&vet=1&usg=AI4_-kTD3Aok1MvKRPc_J1J9NQaeiGHZyQ&sa=X&ved=2ahUKEwipx4f0_7zvAhWnAGMBHev1AqUQ9QF6BAgYEAE&biw=1707&bih=821&dpr=1.13#imgrc=AO-3uctMi1z2RM

When did what Malthus’ predicated as “the point of crisis” occur? Use your calculator.

4. Actually, the following table shows real population growth in England during nineteenth century. How can you explain such a difference between Malthus’ theory and reality?

Year 1800 1820 1840 1860 1880 1900

Population (million) 10 15.5 20.2 24.5 33 41.5

Answer 1

1a)P(n) = 10(1.03) P(n-1) or as a function P(n) = 10(1.03)ⁿ, where n is the number of years since 1800 and P(n) is the population in millions of people. It's an exponential function? I'm not sure what exactly they're looking for here.

b) The population estimate for 1830 is given as P(30) = 10 (1.03)³⁰≈ 24.27 million people

The population estimate for 1850 is given as P(50) = 10 (1.03)⁵⁰≈ 43.84 million people

c) This is just asking us to calculate the n for which P(n) = 30

So 30 = 10(1.03)ⁿ

3 = 1.03ⁿ

Solving this for n, we get n ≈ 37.17, so in the year 1837, it would surpass 30 million according to his estimate.

2a) R(n) = 0.48 + R(n-1), where R(0) = 12. This is a linear sequence. (As a function it is R(n) = 12 + 0.48n, where n is the number of years since 1800)

b) This is simply asking us for R(30) and R(50). R(30) = 12 + 0.48(30) = 26.4 million people and R(50) = 12 + 0.48(50) = 36 million people.

c) Similarly to this problem, this is asking for what n, does R(n) = 30.

30 = 12 + 0.48n

18 = 0.48n

1800 = 48n

1800/48 = n

150 / 4 = n, so 75/2 = 37.5 = n

In this case, it will reach 30 million in the year 1837, but slightly later in the year.

3) To find this point, we simply need to find the point where the functions are equal:

10*(1.03)ⁿ = 12 + 0.48n

1.03ⁿ = 1.2 + 0.048n

If we solve for n, we get that n ≈ 36.77, so in the year 1836 the point of crisis occurs.

4) It could be that he got the growth constant incorrect. Or simply the fact he didn't take into account things like famines and scarcity of resources and birthrates and the like affect population growth in a somewhat random fashion.