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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.

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

Probably need use:

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

https://brainmass.com/math/consumer-mathematics/proportionate-growth-rate-equilibrium-size-541659

#### 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.

\$2.19

## If the price elasticity of demand for cable TV connections is high and the price elasticity of demand for movies shown in theatres is less than 1, what strategy would you expect cable TV firms to follow in arranging for initial connection?

Please see attached and explain with clear reasoning and use diagrams where appropriate. Thank you.

Question 1:
a) If the price elasticity of demand for cable TV connections is high (for example greater than1.5) and the price elasticity of demand for movies shown in theatres is less than 1, what strategy would you expect cable TV firms to follow in arranging for initial connection?

b) The Stop Decay Company sells an electric toothbrush for \$25. Its sales have averaged 8,000 units per month over the last year. Recently, its closest competitor Decay fighter, reduced the price of its electric toothbrush from \$35 to \$30. As a result, Stop Decay's sales declined by 1,500 units per month.
(i) What is the arc cross price elasticity of demand between Stop decay's toothbrush and Decay fighter's toothbrush?
What does this indicate about the relationship between the two products?
(ii) If Stop decay knows that the arc price elasticity of demand for its toothbrush is -1.5, what price would Stop decay have to charge to sell the same number of units as it did before the Decay fighter price cut? Assume that Decay fighter holds the price of its toothbrush constant at \$30.
(iii) What is Stop Decay's average monthly total revenue from the sale of electric toothbrushes before and after the price change determined in part (ii) ?
(iv) Is the result in part (iii) necessarily desirable? What other factors would have to be taken into consideration?

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.
b) Suppose that due to changing tastes there is a sudden increase in demand for emu meat
(i) Assuming the costs of emu meat production remain unchanged, what would you expect to happen to profits in the emu meat industry?
(ii) Assuming that there are low barriers to entry into the emu meat industry, explain why resources would move into this industry.
(iii) In the long run, what would you expect to happen to profits in this industry?
(iv) Will normal profits occur in this industry in the long run? If so, explain why.

c) Critically analyse and explain the following statements:
(i) "Wants aren't insatiable. I can prove it. I get all the coffee I want to drink every morning at breakfast"
(ii) "Goods and services are scarce because resources are scarce".

Question 3:
(a) During a five year period, the ticket sales of a city's professional soccer team have increased by 30 percent at the same time that average ticket prices have risen by 50 percent. Do these changes imply an upward sloping demand curve? Explain.
(b) Explain and illustrate graphically the effect of:
(i) An increase in income on the demand curve of an inferior good.
(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.

(c) If the price of VCR's declines by 20 percent and the total revenue from the sale of VCR's rises, what can you say about the price elasticity of demand for VCR's? Will this price reduction necessarily lead to an increase in profits for the VCR manufacturers?

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
(ii) Keynesian economics
(iii) Which model you do think is more realistic?
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:
(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.
(ii) A department store assistant
(iii) A car assembly line worker
(iv) A heavily indebted farmer
(v) A retired business person whose current income is from interest on government bonds.
(vi) The owner of an independent corner shop

(b)Explain and illustrate
(i) "Policies that stimulate the domestic economy tend to cause a trade deficit"
(ii) "A deficit on the balance on current account will impose severe constraints upon domestic economic policies"
(c )Explain how an increase in aggregate demand that increase real domestic output towards the full employment level may generate instability in the price level. What is this effect referred to us?

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