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Proportionate growth rate & equilibrium size

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A population of beavers was introduced into a reserve on 1 January in a particular year, and the size of the population was estimated on the same date in each subsequent year. The size of the initial population was 100, and it had grown to approximately 180 after one year. After one further year, the size of the population was approximately 288. Assume that the behavior of this population satisfies the logistic model.

Show that the annual proportionate growth rate for the population of size 100 was approximately 0.8, and that the annual proportionate growth rate for the population of size 180 was 0.6.

Find the corresponding exact values of the annual proportionate growth rate for low population levels r, and the equilibrium population size E.

Answer to (a)
Year # 1 = 100 (P0)
Year # 2 = 180 (P1)
Year # 3 = 288 (P2)

Pn = (1 - r)nP0 (n=0,1,2,...)

P1 = (1 + r)1100 P2 = (1 + r)nP1
180 = (1 - r)1100 288 = (1 + r)n 180
1 + r = 180/100 1 + r = 288/180
r = 180/100 - 1 r = 288/180 - 1
r = 0.8 r = 0.6

Answer to (b)

Probably need use:

Pn+1 - Pn = rPn ( 1 - Pn/E )

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Solution Preview

Initial population, Po = 100.
Let us assume population growth rate, r1 = 80% == 0.80

Hence, population after one year,
P1 = Po + r1*Po
= Po*(1+r1)
= 100*(1+0.8)
= 100*1.8 = 180 == given number --Proved

For next year,
Initial population, Po = 180
Let us ...

Solution Summary

We solve a problem related to population growth rate different two different years, followed by estimation of equilibrium population size & proportionate growth rate.

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Question 1
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(i) Classical economics
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(c) "Unemployment can be avoided as long as businesses are willing to accept lower product prices, and workers to accept lower wage rates." Critically evaluate this statement.
(d) Explain and evaluate the following statements in terms of Keynesian- monetarist controversy:
(i) "If the national goal is to raise income, it can be achieved only by raising the money supply".
"The size of a government's budget deficit is not important. What is important is how the deficit is financed".

Question 2

(a)Evaluate as accurately as you can the manner in which each of the following individuals would be affected by fairly rapid inflation.
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