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Hi, I need help on a math problem set containing 3 questions that have 5-7 parts to them (a,b,c…). It’s due pretty urgently so if you could get it done quickly that would be greatly appreciated.
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Math Tools/Math for Econ
Assignment # 8
1. Suppose there are a fixed number of employees of a company, spread between two branch
offices in Abbotsford and Burnaby. Suppose each month 4/5 of Abbotsford employees are
transferred to Burnaby and ½ of Burnaby employees are transferred to Abbotsford. Suppose
currently there are 100 and 120 employees in Abbotsford and Burnaby respectively.
a) Find a system of difference equations for this problem. Convert this to a matrix equation.
b) Use matrix multiplication to determine the number of employees after 1 month.
c) Use an online matrix calculator to find the eigenvalues. Confirm that the dominant eigenvalue
is 1. (Why must it be 1?)
d) Find the eigenvectors corresponding to each eigenvalue.
e) Express the initial values as a linear combination of the eigenvectors. You can use an online
calculator to solve for c1 and c2 as we did in class.
f) Use your expression in (e) to write an expression for the number of Abbotsford and Burnaby
employees after n months using the eigenvalues and eigenvectors.
g) What happens to the distribution of employees in the long term? Explain?
2. Bison can be broken into 3 distinct age groups: calves (under 1 year old), yearlings (1-2
years), and adults (over 2). Studies indicate that about 60% of the calves survive to become
yearlings; about 75% of yearlings survive to become adults; 5% of adults die each year. On the
average, about 42 calves are born each year for every 100 adults.
a) Find a system of difference equations for these age classes. Convert this to a matrix equation.
b) If this year there are 100 calves, 50 yearlings and 250 adults, how many will there be next
year? (Use matrix multiplication.)
c) Use an online matrix calculator to find all the eigenvalues. Determine the dominant
eigenvalue. What does it tell you about the long-term population levels of the bison?
d) Find the eigenvector corresponding to the dominant eigenvalue. What does it tell you about
the long-term age distribution of the bison?
3. Suppose the population of a town decreases by 3% a year and also has 100 people leave each
year.
a) Determine an appropriate difference equation for the problem.
b) Sketch a graph of change of population versus population.
c) Use your sketch in (b) to determine the stability of the equilibrium.
d) Find the general solution to the difference equation and sketch several time plots.
e) Find the definite solution if in the year 2014 there were 1500 people in the town.
f) Use your answer in (e) to determine when the population will double.
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