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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.

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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