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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.

(a)
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.

(b)
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

a.
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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c) "Because of the Australian love affair with driving motor vehicles, increases in the price of petrol will not affect consumption". What type of demand curve is implied by this statement? Do you believe this is true? Why?.

Question 2
a) Explain the differences between accounting profit, economic profit and normal profit.
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c) Critically analyse and explain the following statements:
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Question 3:
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(ii) A drop in the price of product L on the demand for substitute product M.
(iii) A decline in income upon the demand curve of a normal good.
(iv) An increase in the price of product J upon the demand for complementary good K.

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MACROECONOMICS:

Question 1
(a) Use the aggregate demand - aggregate supply model to explain the consequences in terms of price - level and real output consequences of a decline in aggregate demand as envisioned by:

(i) Classical economics
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Compare classical and Keynesian interpretations of the aggregate demand curve.
(b) Show the implications of the supply siders' arguments in the aggregate demand- aggregate supply framework. Do you think that tax-transfer disincentives have a short-run or long -run impact or both? Explain.
(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:
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Question 2

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(vi) The owner of an independent corner shop

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