Fox Studios is a pest rabbit control company commissioned to assist with a wild rabbit population growth problem on a small island. 1200 rabbits were introduced onto an island, but it was soon realised that the population growth of the rabbits would cause environmental problems on the island. Fox Studios was commissioned to assist with this problem by introducing foxes who help contain the rabbit numbers. The rabbit population growth rate is 10% each month, (This is the net of the natural birth rate and death rate). Each fox eats 10 rabbits per month. (The foxes are also fed by the company as part of their cost). (a) Show that R₁+1 = 1.1 R₁ - 12 F where R, is the rabbit population at month t and Fis the number of foxes employed, is a suitable difference equation to use. (b) Verify that R₁ = (1200 - 120 F) × 1.1 + 120 F is the solution to the difference equation. (c) Fox Studios is faced with the following optimisation problem: i. The contract is for 5 years (60 months) ii. Each fox costs $350 000 (special training, transportation, feeding and medical expenses over the 5 years) iii. The company is liable to pay a fine of $1 per rabbit for any excess over 20 000 in every month during the 5 year life. Solve for the number of foxes that should be sent to minimise the cost to the company. Hints: i. ii. iii. There are 11 cases to consider F=0, 1, 2, ... 10 R is an increasing function in t with a calculable time of first breach say T*. An EXCEL spread sheet may be used, but theoretical calculations should be performed.
Fox Studios is a pest rabbit control company commissioned to assist with a wild rabbit population growth problem on a small island. 1200 rabbits were introduced onto an island, but it was soon realised that the population growth of the rabbits would cause environmental problems on the island. Fox Studios was commissioned to assist with this problem by introducing foxes who help contain the rabbit numbers. The rabbit population growth rate is 10% each month, (This is the net of the natural birth rate and death rate). Each fox eats 10 rabbits per month. (The foxes are also fed by the company as part of their cost). (a) Show that R₁+1 = 1.1 R₁ - 12 F where R, is the rabbit population at month t and Fis the number of foxes employed, is a suitable difference equation to use. (b) Verify that R₁ = (1200 - 120 F) × 1.1 + 120 F is the solution to the difference equation. (c) Fox Studios is faced with the following optimisation problem: i. The contract is for 5 years (60 months) ii. Each fox costs $350 000 (special training, transportation, feeding and medical expenses over the 5 years) iii. The company is liable to pay a fine of $1 per rabbit for any excess over 20 000 in every month during the 5 year life. Solve for the number of foxes that should be sent to minimise the cost to the company. Hints: i. ii. iii. There are 11 cases to consider F=0, 1, 2, ... 10 R is an increasing function in t with a calculable time of first breach say T*. An EXCEL spread sheet may be used, but theoretical calculations should be performed.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Transcribed Image Text:Fox Studios is a pest rabbit control company commissioned to assist with a wild rabbit population
growth problem on a small island.
1200 rabbits were introduced onto an island, but it was soon realised that the population growth of
the rabbits would cause environmental problems on the island. Fox Studios was commissioned to
assist with this problem by introducing foxes who help contain the rabbit numbers.
The rabbit population growth rate is 10% each month, (This is the net of the natural birth rate and
death rate).
Each fox eats 10 rabbits per month. (The foxes are also fed by the company as part of their cost).
(a) Show that R₁+1 = 1.1 R₁ - 12 F where R, is the rabbit population at month t and F is
the number of foxes employed, is a suitable difference equation to use.
(b) Verify that Re = (1200 120 F) x 1.1 + 120 F is the solution to the difference
equation.
(c) Fox Studios is faced with the following optimisation problem:
i. The contract is for 5 years (60 months)
ii.
Each fox costs $350 000 (special training, transportation, feeding and
medical expenses over the 5 years)
iii.
The company is liable to pay a fine of $1 per rabbit for any excess over
20 000 in every month during the 5 year life.
Solve for the number of foxes that should be sent to minimise the cost to the company.
Hints:
i.
ii.
iii.
There are 11 cases to consider F=0, 1, 2, ... 10
R is an increasing function in t with a calculable time of first breach say T*.
An EXCEL spread sheet may be used, but theoretical calculations should be performed.
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