数学代写-2160A
时间:2020-12-13
Econ 2160A-001 E. Rivers
Ch. 1-3 1
CHAPTERS 1-3: The portions of these chapters that we are covering provide the legal
background and framework for the taxation system in Canada, some basics of the theory of
welfare economics, and some important mathematical tools that will help us analyze situations
that involve costs and benefits that may be incurred at different periods of time. These
concepts will underlie material in a number of different chapters and topics as we proceed
through the semester.

CHAPTER 1: INTRODUCTION TO PUBLIC FINANCE IN CANADA
Canada’s Government at a Glance: The Legal Framework
Different levels of government have differing legal authority to tax.
1. Federal Government
• Power to raise money by any system or mode of taxations (except on provincial lands and
property)
• Responsibilities include: national defence, navigation and shipping, regulation of trade and
commerce, the criminal justice system, and money and banking, unemployment insurance,
and subsidies to provinces.
2. Provincial Governments
• Limited to use of direct taxes
• Forbidden from use of indirect taxes except on natural resources (nonrenewable and
forestry)
o Provinces now collect sales tax and excise taxes on gas, alcohol, and tobacco. These
would normally be classified by economists as indirect taxes, and thus wouldn’t be
allowed at the provincial level. However, these have been classified as direct taxes
by the government in this context, essentially by interpreting the merchant as a
collection agent for a tax imposed on end consumers of the goods and services in
question.
• Some provinces receive funds in the form of transfers from federal government.
• Responsibilities include: health, education, welfare, transfers to local governments
3. Local Governments
• Have only taxing and spending powers that the provincial governments choose to delegate
to them.
• Dominant revenue source is the property tax. Local governments also receive transfers
from the provincial government.
• Responsibilities include: public safety, sanitation, local infrastructure
Note: There is movement of areas of responsibility between the levels of government from time to time.
With those changes, there are often agreements about how revenue will get to the level of government
providing that service. So there can be changes in taxation and subsidies over time as well.
Econ 2160A-001 E. Rivers
Ch. 1-3 2
CHAPTER 2: FUNDAMENTALS OF WELFARE ECONOMICS (Presented in the context of the
Chapter 3 discussion of the efficiency of the competitive equilibrium and the Chapter 3
appendix on consumer and producer surplus.)
Welfare economics: Concerned with the social desirability of alternative economic states.
• The term welfare is used in this context to mean wellbeing or happiness
Pareto Efficiency: an allocation of resources such that no person can be made better off
without making another person worse off.
Pareto Improvement: a reallocation of resources that makes at least one person better off
without making anyone else worse off
• Let’s consider the (pareto) efficiency of a competitive equilibrium using a figure from
Chapter 3 (Figure 3.1)
• Recall:
o Marginal benefit (MB) of consumers reflected by maximum willingness to pay for
each unit. This is the basis for the demand curve.
o Marginal cost (MC) of producers reflected by minimum supply price for each
unit. This is the basis for the supply curve.

• Ignore what you know about where we will end up in a competitive equilibrium for a
moment. Consider any arbitrary starting point…
o At a point like Q1: MB exceeds MC at this quantity. Consumers would be better
off if they could consume more apples. If consumers pay any price between MB
and MC of the next apple, then at least one group and possibly both groups are
better off.
Ø This would be a pareto improvement. Thus, Q1 was not an efficient point.
Econ 2160A-001 E. Rivers
Ch. 1-3 3
o At a point like Q2: MC exceeds MB at this quantity. If we are actually trading this
quantity, producers are taking a loss on the last apple(s) sold. The cost savings
from reducing the quantity traded is more than enough to compensate
consumers for the reduction in apples consumed. With the right payment from
producers to consumers, at least one group and possibly both groups would be
better off.
Ø This would be a pareto improvement. Thus, Q1 was not an efficient point.
o At Q*, no pareto improvement is possible. This is the only point where no pareto
improvement is possible.
Ø This is the competitive equilibrium. The competitive equilibrium is (pareto)
efficient.
• We can assess and compare changes in welfare at the market level by looking at
changes in consumer and producer surplus
o Example on the consumers’ side:

o Example on the producers’ side:

Econ 2160A-001 E. Rivers
Ch. 1-3 4
Important Theorems in Welfare Economics:
• The First Fundamental Theorem of Welfare Economics states that a Pareto Efficient
allocation of resources emerges
o A competitive economy “automatically” allocates resources efficiently without
any need for centralized direction
o Necessary assumptions:
§ All producers and consumers act as perfect competitors; no one has any
market power
§ A market exists for each and every commodity
• The Second Fundamental Theory of Welfare Economics states that society can attain
any Pareto efficient allocation of resources – including one that is more equitable – by
making a suitable assignment of initial endowments and then letting people freely trade
with each other
o i.e., Equity (fairness) can be achieved without inhibiting efficiency
o This is a very important point in our discussion, because in real life, society
(government) does NOT choose the allocation. Individuals earn it. However, the
taxation system allows a mechanism for redistributing that allocation!
o Some caveats:
§ The redistribution mechanism (in most cases income taxes) itself induces
inefficiencies, so not quite as good as if we could choose the correct
initial endowments.
§ The assumptions of competitive markets and formal markets for all goods
do not always hold.
– Market power
• Example: Monopoly
– Nonexistence of markets
• Externalities (a cost or benefit is generated to someone
other than the producer or consumer of a good, so that
cost or benefit is not incorporated into its market price)
• Public Goods (Public goods are a key example of why
taxation is necessary. Tax revenue is used to pay for
things we deem to be necessary or beneficial to our
society that would not be produced or would not be
produced in efficient quantities if the government did not
step in and coordinate the provision of these goods and
services.)
• Asymmetric Information (one party has information about
a good that the other does not)
§ We may care about some people more than others.
– “Distributional considerations”: We may deem some individuals
more deserving than others when it comes to income
redistribution or public goods provision.
Econ 2160A-001 E. Rivers
Ch. 1-3 5
– Can be addressed with
• concept of “potential pareto improvement”;
• or by assuming that a dollar benefit to one group is more
than a dollar going to a different group (see “distributional
weight” below).
CHAPTER 3 COST-BENEFIT ANALYSIS
Evaluating projects:
• Net benefit of a project = Benefit – Cost
• Two possible projects: Project 1, Project 2
• Two groups that incur costs and enjoy benefits of each project: Group A, Group B
• Notation example: BenefitA1 denotes the benefit to Group A of Project 1.
o Net Benefit1 = (BenefitA1 + BenefitB1) – (CostA1 + CostB1)
o Net Benefit2 = (BenefitA2 + BenefitB2) – (CostA2 + CostB2)
• Is Net Benefit1 > < or = Net Benefit2?
o If we have no reason to value the welfare or change in welfare of the individuals
in Group A over those in Group B, then we would simply compute the net
benefits and compare.
o But what if we do care about the groups differently?
§ “Distributional Weight” (w) on Group B: How much we value a dollar
spent or enjoyed by someone in Group B relative to $1 spent or enjoyed
by someone in Group A.
§ Net Benefit1 = (BenefitA1 + w*BenefitB1) – (CostA1 + w*CostB1)
§ Net Benefit2 = (BenefitA2 + w*BenefitB2) – (CostA2 + w*CostB2)
§ Two types of questions we typically ask:
1. We know w. Which project has higher net benefit?
2. What value of w makes the net benefit of the two projects
equal?
« I posted an example of this type of question with the course
materials for Chapter 1-3 for you to try. I also posted a video
that works though the example step by step.

The importance of timing:
• If costs or benefits come at different times, this can complication the evaluation or
comparison of the net benefit of projects. The value of $1 changes over time, so we
must incorporate this into our analysis. To make correct comparisons, all dollars must
be in the same timeframe.
• Projecting Present Dollars into the Future
o Notation:
§ R = the dollar value of a lump sum cost or benefit
§ T = the number of time periods
Econ 2160A-001 E. Rivers
Ch. 1-3 6
• we will typically measure this in years in this course
• period 0 = now
§ r = the percentage interest rate corresponding to that frequency of time
period
o Assumptions:
§ No default risk
§ No inflation
o How much will $R today be worth in T years at interest rate r?
§ Future value of R is R(1+r)T
o = (1 + )!
• Discounting Future Dollars to the Present
o In the example above, R was the present value (PV) and we were computing the
future value (FV) and we had = (1 + )!
o Therefore, = "#(%&')!
o Terminology:
§ r is the “discount rate”
§ (1+r) is the “discount factor”
§ If r is low, this reflects that people are patient, more future-oriented
• Favours projects with returns farther into the future
§ If r is high, this reflects that people are impatient, more present-oriented,
“live in the now”
• Favours projects with more immediate returns
• Discounting a Stream of Income
o What if you will pay/receive $R0 in now (period 0), $R1 in period 1, etc. until
some specific period T?
o You can compute the present value of each R separately and add them together. = ) + %(1 + ) + *(1 + )* +⋯+ !(1 + )!
o What if the payments continue indefinitely (“in perpetuity”)?
§ For simplicity, let’s assume that R is the same in each period
(R=R0=R1=R2=…) = + (1 + ) + (1 + )* + (1 + )+ +⋯
§ This is equivalent to: = +
§ If the payments do not begin until period 1 (one year from now), this is: =
§ Why? Let’s use the example with payments beginning in period 1. Our
starting equation (Equation A) is: = (1 + ) + (1 + )* + (1 + )+ +⋯
Econ 2160A-001 E. Rivers
Ch. 1-3 7
• Will now do a little “math trick” to do this. I expect you to
understand that what we are doing is mathematically correct, but
I would not expect you to come up with this on your own or
recreate it. We will divide both sides by (1=r). This gives us
Equation B: (1 + ) = (1 + )* + (1 + )+ + (1 + ), +⋯
• Notice that the right-hand side of Equation B is the same as the
portion of the right-hand side of Equation A after the first term. I
have highlighted these in blue font in the above equations.
• We can substitute Equation B into that portion of Equation A to
get: = (1 + ) + (1 + )
• Now simply solve for PV: (1 + ) = + (1 + ) − = + − = = =
• If you want the version of this present value that includes a
benefit or payment in the current period, you can simply add it, as
the current period amount is already in present value terms. = +
o What if we relax the assumption we made earlier of no inflation?
§ We will need to account for changing prices
§ Example: A public works project that provides ongoing benefit from a
public good or service provided over a period of time. The government
borrows to finance the cost of the project and will have to pay back with
interest over a period of time.
§ R needs to be adjusted for inflation (π used to denote the percentage of
inflation) to arrive at value of the benefit in the year the benefit is
received
• R is the value of the benefit now
• R(1+π) is the value of the benefit one year from now
• R(1+π)(1+π) or R(1+π)2 is the value of the benefit two years from
now
• R(1+π)T is the value of the benefit T years from now
§ The discount factor (1+r) also needs to be adjusted for inflation, as
lenders know that loans will be paid back with discounted dollars.
• Nothing needs to be done in period 0 (no discounting necessary in
period 0)
Econ 2160A-001 E. Rivers
Ch. 1-3 8
• (1+r) is the year 1 discount factor. It becomes (1+r)(1+π) when
adjusted for inflation.
• (1+r)2 is the year 2 discount factor. It becomes (1+r)2(1+π)2 or
((1+r)(1+π))2 when adjusted for inflation.
• (1+r)T is the year T discount factor. It becomes (1+r)T(1+π)T or
((1+r)(1+π))T when adjusted for inflation.
§ The PV calculation (let’s assume the we are in the “in perpetuity”
example) is now: = + (1 + )(1 + )(1 + ) + (1 + )*(1 + )*(1 + )* + (1 + )+(1 + )+(1 + )+ +⋯
§ The inflation adjustment in each term’s numerator cancels the inflation
adjustment in each term’s denominator (as long as both streams R and r
are adjusted for inflation)! We are left with: = + (1 + ) + (1 + )* + (1 + )+ +⋯
§ As explained above, this will equal: = +
§ What if R is a fixed, nominal dollar amount (so R does not have to be
adjusted for inflation), but the benefit is still financed through
borrowing?
• R no longer has to be adjusted for inflation, but the discount
factor still does!
• In this case, we will get: = + (1 + )(1 + ) + ((1 + )(1 + ))* + ((1 + )(1 + ))+ +⋯
• Note that any term (1+r)(1+π) can be expanded to (1+r+rπ+π) and
then rewritten as (1+(r+rπ+π)).
• It follows that this present value will equal = + -(.&./&/).
Note that for typical interest rates and typical inflation rates, the
product rπ is extremely small, so we often use r+π as a good
approximation of r+rπ+π, and this present value approximation
becomes = + -(.&/).
Combining project evaluation with timing considerations:
• In the example above, we computed the present value of a stream of revenues OR a
stream of costs. Typically, we would want to compute the present value of a stream of
net benefits (benefit-cost). We can do this by simply combining the concept of net
benefit (benefit-cost) with the mathematical process of computing present value.
Rather than computing the present value of an individual amount R, we can replace R
with (B – C) where B represents the benefit and C represents the cost. From there, the
math of computing present value is the same.
Econ 2160A-001 E. Rivers
Ch. 1-3 9
o The internal rate of return is the discount rate that would make the present
value of the project just equal zero.
§ A simple example: A project costs $1,000 today and generates a benefit
of $1,200 one year from now. What is the internal rate of return? In
other words, what is the discount rate that makes $1000 today equal to
$1,200 one year from now? The internal rate of return is 20% (because
$1000*(1+20%)=$1,200).
• In some settings, analysts prefer to compute a benefit-cost ratio rather than present
value of the net benefit. To do so, we compute B/C where B is the present value of the
benefit in all time periods and C is the present value of the costs in all time periods.

































































































































































































































































































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