2. (15 points) For the tal For the rational functions below, find all zeros, holes, vertical
asymptotes, and the long-run havior. If the function does not have a vertical asymptote or hole
please indicate so. (a) f(1) = Zeros: Vertical Asymptotes: Holes: Long-Run Behavior: (x+1)(x +
3) (b) f(x) =- - 4x(+ 3) Zeros: Vertical Asymptotes: Holes: Long-Run Behavior:
Solution
a) f(x) = ( x^2 - 9 ) / x^3
zeros are
x^2 - 9 = 0
x^2 = 9
x = + - 3
zeros ( 3,0) and (-3,0)
vertical asymptotes are
x = 0
holes - none
long run behaviour
f(x) = x^2 / x^3 = 1/x
so , as x tends to + - infinity f(x) will tend to 1/x
b) f(x) = ( x+1)^2 ( x+3) / - 4x ( x+3)
zeros are
( x+1)^2 = 0
x = -1
zero is (-1,0)
vertical asymptotes are
x = 0
holes are x = -3
long run behaviour f(x) = x^3 / -4x^2
f(x) = -x /4
the graph will behave like f(x) = -x/4 for larger values.
HMCS Vancouver Pre-Deployment Brief - May 2024 (Web Version).pptx
2. The Lorenz curve measures inequality in person income distribution.pdf
1. 2. The Lorenz curve measures inequality in person income distribution. What is the difference
between personal income distribution and functional income distribution? 3. What is the
difference between a proportional tax and a progressive tax? What are the problems of
progressive taxation as a means to achieve greater equality in income distribution?
Solution
Distribution - Division of economy within an economy
Personal income distribution - Division amongst individuals. It is the income share accruing to
factors of individuals. It is a microeconomic phenomenon. Frameworks have been developed by
micro-economists for exploring factors shaping the distribution of income across individuals.
Theoretical analysis is dominated by the neoclassical perspective.
Functional income distribution - Division between groups of people. Regardless of income
sources, it is the distribution of income across individuals. It is a macroeconomic issue.
Macroeconomists construct models that capture the determinants of the functional income
distribution.
Interaction can be explained by stock-flow consistent macroeconomic models. Labour, profits,
rent, interest, and, unemployment benefits are five sources of income. Gini coefficient, and,
squared coefficient of variation are the measures of inequality that capture personal income
distribution. Inequality of total income is closer to inequality of income from labour.
Declining labour income is associated with growing labour income. A percentage decrease in
labour share of income increases the Gini coefficient. Developments in the distribution of wages
are key factors in the dynamics of inequality. Some of the reasons for labour inequality are
public revenues, public health spending, labour share, economic globalization, and, public social
protection spending.
2. Proportional, and, progressive tax
Progressive tax - Higher percentage from higher income groups. For instance, income tax slabs
Proportional tax - the same percentage of tax from all income groups. For instance, sales tax. It
can be considered regressive as a larger proportion is income is taken from the incomes of those
with a lower income.
Income taxes are progressive but providing a larger exemption in the budget can lead to a higher
budget deficit. A reduction in Government spending impacts other social welfare benefits. Social
security, and, property taxes are regressive. Income taxes are progressive. This results in a
proportional system for taxpayers. The taxpayers do not gain that much from the progressive
income tax system.