Ch1
Cost and Benefit Analysis
Dr. Si Chen
University of Hong Kong
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Table of Contents
• Motivation
• Cost and Benefit Analysis
• Opportunity Cost
• Inference, Prediction and Policy
• Optimal Quantity
• Allocation of Resources
• The Invisible Hand
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Motivation
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Will you pick up the check?
• There is an opportunity for you to pick up a check of $500 at
Disneyland. The ticket to enter Disneyland costs $400.
• Yes?
• No?
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Will you pick up the check?
• There is an opportunity for you to pick up a check of $100 at
Disneyland. The ticket to enter Disneyland costs $400.
• Yes?
• No?
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Cost-Benefit Analysis in Everyday Life
• An individual (or a firm or a society) often make decisions based
on the comparison between the benefits and costs of the options.
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Cost-benefit analysis
Understanding Decision-Making in Economics
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The Cost-Benefit Principle
The Cost-Benefit Principle
A rational individual, firm, or society should take an action if, and
only if, the extra benefits are at least as great as the extra costs.
• Formula:
• B(x) ≥ C(x)
• Where:
• B(x) = Benefits of action x
• C(x) = Costs of action x
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Example – Class Size Decision
• Option 1: 1 class of 300 students
• Cost: $300 per student
• Option 2: 12 classes of 25 students
• Cost: $3,600 per student
• Trade-off:
• Smaller classes = better learning outcomes but higher costs.
• Decision:
• Choose based on whether the benefit (better learning) justifies the cost.
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The Scarcity Principle
• Key Idea: Resources are limited, but wants are unlimited.
• Implication: Every choice involves trade-offs.
• Example:
• Rich or poor, everyone has 24 hours a day – time is a scarce resource.
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Rationality in Economics
Rational
An individual is said to be rational if the person has well-defined
goals and thus try to fulfill them as best as possible.
• Examples:
• Firms maximize profit
• Consumers maximize satisfaction from consumption
• Governments maximize …
• Guiding Principle:
• Cost-benefit analysis helps maximize objectives.
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Is rationality a realistic assumption?
• Treat rationality as a working assumption and the cost-benefit
analysis as a guiding principle.
• We should judge the assumption and the principle by its
usefulness in explaining and predicting real world phenomena.
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Estimating Costs and Benefits
• Costs are benefits are not always readily denoted in monetary
units.
• Estimation is often required.
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Example--“Sleep No More”
“Sleep No More adapts the story of Macbeth, deprived
of nearly all spoken dialogue and set primarily in a
dimly-lit, 1930s-era establishment called the McKittrick
Hotel... the audience walks at their own pace through a
variety of theatrically designed rooms… the physical
location, rather than being a traditional playhouse, is an
imitation of the actual setting… the audience may move
through the settings, interact with the props, or observe
the actors at their own pace...”
o Would you like to see it? How badly do you want to see it?
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Willingness to Pay
• A way to quantify your liking is through the concept of
willingness to pay (WTP).
• We ask “What is the maximum I’m willing to pay for it?”
• For example, you are willing to pay maximum $1000 to see the
drama. Then you willingness to pay for the drama is, $1000.
• Note that, WTP is to be distinguish from utility. While the latter
has a much more important role in Economics, the former helps
us to gain a first intuition of economic decision making.
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Estimating Costs and Benefits
• Estimating Benefits: "What is the maximum I’m willing to pay for
this action?"
• Estimating Costs: "What is the minimum benefit required to take
this action?"
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Example -- Turning off an alarm:
• Benefit: Removing annoyance (e.g., $100)
• Cost: Leaving a comfortable bed (e.g., $10)
• Decision:
• If B(x) ≥ C(x), take the action.
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Economic Surplus
Economic Surplus
Economic Surplus = B(x) – C(x)
• Interpretation: The net gain from taking an action.
• Decision Rule: Take action if Economic Surplus ≥ 0.
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Opportunity Cost
Understanding What You Give Up to Choose
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Definition of Opportunity Cost
Opportunity Cost
The value of the next best alternative that you must give up when
making a decision.
• Opportunity cost includes:
• Includes monetary and nonmonetary costs
• Includes explicit and implicit costs.
• Key Idea: Every choice involves a trade-off.
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Example
-- Opportunity Cost of Seeing Eason Chan I
• Free ticket to Eason Chan concert (non-refundable) vs. pay for a
$800 ticket to Yo-Yo Ma concert
• Willing to pay up to $1000 for Yo-Yo Ma.
• Assume there are no other costs or benefits beyond what is given.
• What is the opportunity cost of seeing Eason Chan?
• $1000-$800 (ticket cost) = $200.
• Key takeaway: Opportunity cost is not always explicit; it is the
value of the forgone next best. Value is the benefits net of the
costs.
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Example
-- Opportunity Cost of Seeing Eason Chan II
• Pay for a $100 ticket to Eason Chan concert vs. pay for a $800
ticket to Yo-Yo Ma concert
• Willing to pay up to $1000 for Yo-Yo Ma.
• Assume there are no other costs or benefits beyond what is given.
• What is the opportunity cost of seeing Eason Chan?
• ($1000+$100)-$800 = $300.
• Key takeaway: Opportunity cost is not always explicit; it is the
value of the forgone next best. Value is the benefits net of the
costs.
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Example
-- Opportunity Cost of Seeing Eason Chan III
• You won three free tickets, one to see the concert by Eason Chan,
one to go to Disneyland to pick up a $400 check, and one to go to
Ocean Park to pick up $500 check.
• You cannot resell any of these tickets (ID will be checked at entrance).
• Everything happens the same time. Assume there are no other costs or
benefits.
• Based on this information, what is the opportunity cost of seeing
Eason Chan?
• $500
• Key takeaway: Opportunity cost is the forgone value of the next
best option.
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Example
-- Check at Disneyland I
• You spend $100 on meals every day. There is an opportunity for
you to pick up a check of $450 at Disneyland. The ticket to
Disneyland costs $400. Should you pick up the check?
• Yes
✓
• No
• Do we include the $100 dollars on meals in our cost and benefit
analysis?
• B=450, C=400
✓
• B=450, C=500
• Key takeaway: only include costs incurred due to the activity.
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Example
-- Check at Disneyland II
• You spend $100 on meals every day. There is an opportunity for
you to pick up a check of $300 at Disneyland. The ticket to
Disneyland costs $400. You happened to own a non-refundable
ticket. Should you pick up the check?
• Yes
✓
• No
• Do we include the $400 dollars in our cost and benefit analysis?
✓
• B=300, C=0
• B=300, C=400
• B=300, C=500
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Example
-- Check at Disneyland III
• You spend $100 on meals every day. There is an opportunity for
you to pick up a check of $300 at Disneyland. The ticket to
Disneyland costs $400. You happened to own a fully refundable
ticket. Should you pick up the check?
• Yes
• No
✓
• Do we include the $400 dollars in our cost and benefit analysis?
• B=300, C=0
• B=300, C=400
• B=300, C=500
✓
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Sunk Cost
Sunk Cost
Sunk cost is a cost that is beyond recovery at the moment a decision
must be made.
• Example:
• Non-refundable ticket to a concert → Sunk cost.
• Refundable ticket → Not sunk cost.
• Key Idea: Only costs affected by the decision should be considered.
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Counting Psychic Cost
• Definition: Psychic cost is the stress or discomfort associated with an activity.
• Example:
• Mary cancels a $200 job to attend a concert.
• The concert ticket costs $50.
• She is willing to pay $150 to hire someone else to do the job for her.
• That is, her psychic cost of the job: $150
• Opportunity Cost:
• $50+$50 (net value of job) = $100.
• Key takeaway: Psychic cost is a real cost and should be included.
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Example – Wind-Surfing vs. Work
• Wind-surfing: Worth $500 to you and costs $100.
• RA job: Pays $450, you’d do it for free.
• Decision: RA job or wind-surfing?
What does
“you’d do it for
free” imply?
• B (wind-surfing) = $500
• C (wind-surfing) = $100 + $450 (forgone salary) = $550
• ES (wind-surfing) = B (wind-surfing) - C (wind-surfing) = -$50
• Don’t go wind-surfing.
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Exercise
--Will you help Uncle Paul?
After an exchange in England, you were about to return to Hong
Kong. You noticed that Air Jordan I sneakers were sold at HK$1,800
a pair at Hong Kong stores and about HK$500 in England. You had
been planning to bring back a pair and sell it at about HK$1,300.
Then Uncle Paul asked you to get a pair back for him, and
promised to pay you the cost (HK$500). You value the chance of
helping Uncle Paul at $x. Assume you can only bring 1 pair back
How will your decision to help him depend on x?
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Exercise
--Will you help Uncle Paul?
Imagine you have not yet purchased the sneakers.
• The benefit of helping Uncle Paul?
• Your valuation of helping Uncle Paul
= x + 500 - 500
=x
• The cost of helping Uncle Paul?
• Helping Uncle Paul, you will lose the opportunity of bringing back the sneakers
and reselling them yourself.
• What you could have gotten
= 1300 - 500
= 800
• Benefit – Cost = x - 800
 You should help Uncle Paul if x > 800.
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Exercise
--Will you help Uncle Paul?
Imagine you are already back with the sneakers.
• The benefit of helping Uncle Paul?
• Your valuation of helping Uncle Paul + reimbursement from Uncle Paul
= x + 500
• The cost of helping Uncle Paul?
• Helping Uncle Paul, you will lose the opportunity of reselling the sneakers
yourself.
• What you could have gotten
= 1300
• Benefit – cost = x + 500 - 1300 = x - 800
 You should help Uncle Paul if x > 800.
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Inference, Prediction and Policy
What we can do with cost-benefit analysis
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Inference, Prediction and Policy
• For rational individuals, actions are chosen to maximize some goals,
subject to their respective constraints.
• Prediction:
• Knowing a decision-maker’s goal and constraint, we can predict her
decision.
• Inference:
• By observing their choice of actions, we can guess the goals and
constraints facing decision makers.
• Policy:
• If we can change their goals/preferences, and/or constraints, we can cause
decision makers to choose a different action.
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Prediction: Will you go to the drama?
• You are willing to pay and can pay $1200 to see “Sleep No More”.
The price of the ticket is $800, and there is no other costs. Will
you go to the drama?
• Yes?
• No?
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Prediction: Will you go to the drama?
• You are willing to pay $500 to see “Sleep No More”. The price of
the ticket is $800, and there is no other costs. Will you go to the
drama?
• Yes?
• No?
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Inference
• Observed action:
• John pays a price of $800 to see the drama.
• Conclusion:
• John is willing to pay more than $800 to see the drama.
• John has more than $800 in his bank account.
• What if we observe that Brandon did not go to the drama? What
conclusions can we draw?
• Brandon is either unwilling to pay $800 to see the drama OR
• He does not have $800 in his bank account.
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Prediction
• If the price was $700, will Jon go?
• What if the price was $900?
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Policy
• If we want more people to see the drama,
• we can advertise it.
• reduce the price of the tickets.
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Example
– Garment and textile industry
• In the 60s to 80s, garment and textile industry in Hong Kong hired
a lot of workers. If you were the boss of a garment and textile
company, how would you pay your workers?
A) By hours of work
B) By pieces of completed garment items
• Key takeaway: align the incentives with your goal.
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Optimal quantity
When the decision is no longer 0 or 1
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Understanding Optimal Quantity Decisions
• Next, we will learn how to determine the optimal quantity of a
good or service using cost-benefit analysis.
• The question is no longer binary (i.e., yes or no).
• Example Question: How many scoops of ice cream should one
consume?
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Example -- Ice-cream
• What’s your favourite ice-cream?
• Say each scoop of ice-cream costs $10.
• How many scoops would you have?
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Example -- Ice-cream
• You can consider the problem as a decision among many options:
1 scoop, 2 scoops, 3 scoops etc.
• We know the cost of these options.
• What about the benefit?
• How much would you be willing to pay for 1 scoop of it?
• How much would you be willing to pay for 2 scoops?
• How about 3 scoops?
• Your answer is perhaps not linear.
• That is, you do not like each scoop of ice-cream equally much.
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Example -- Ice-cream
• TB (n units) = the total benefit of having n scoops
• TC (n units) = the total cost of having n scoops
• Calculate ES(n units) = TB(n units) - TC(n units) for every n.
• Pick the n which maximizes the economic surplus.
• Is there an easier way to decide?
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Example -- Ice-cream
• The key is to convert the original decision to a sequence of “YesNo” decisions!
• Start from zero scoop. Should I have the first scoop?
• Start from one scoop. Should I have the second scoop?
• …
• Start from (n-1)-th scoop. Should I have the n-th scoop?
• I would have the n-th scoop if the additional benefit of the n-th
scoop is larger than additional cost of the n-th scoop.
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Thinking at the Margin
• We can think of the trade-off at the margin. Marginal means
additional.
• We consider the marginal cost and marginal benefit when making
a decision.
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Discrete Quantity Case
• MB(n-th unit): the additional benefit due to the n-th unit.
• TB(n units) – TB(n-1 units)
• MC(n-th unit): the additional cost due to the n-th unit.
• TC(n units) – TC(n-1 units)
• Should I do the n-th unit of activity X?
• Yes, if MB(n-th unit) > MC(n-th unit)
• Stop if the opposite.
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Example
-- Ice-cream
Scoop
TB
MB
MC
Consume?
1
19
19
10
Yes
2
28
9
10
No
3
27
-1
10
No
4
16
-11
10
No
• Optimal quantity: 1
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Continuous Quantity Case
• Some goods or activities are not discrete, instead the are perfectly
divisible, or, continuous.
• Think about your working time on polishing an essay.
• How much time should you spend on the essay?
• We want to make an accurate decision. So we have to think about
very small margins.
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Example -- Ice-cream (continuous)
• Now imagine, you can consume any
amount of ice-cream.
• Each little bit give you some benefit and
has some cost.
• Assume
• MB(Q) = 29 - 10Q.
• MC(Q) = $10 per scoop.
• Optimal Quantity: Find Q where
MB(Q) = MC(Q).
• 29 - 10Q = 10 ⇒ Q* = 1.9 scoops.
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Example -- Ice-cream (continuous)
Optimality Check:
• At Q < Q (e.g., Q = 1.8):
• MB(1.8) = 11 > MC(1.8) = 10 ⇒ Expand consumption.
• At Q > Q (e.g., Q = 2.0):
• MB(2.0) = 9 < MC(2.0) = 10 ⇒ Reduce consumption.
• Conclusion: Q* = 1.9 is the optimal quantity.
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Understanding Marginalism
in Continuous Case
• In continuous case, we are applying the same marginal principle
as in the discrete case.
• You should carry on an activity if an only if
TB(Q+ D) - TB(Q) > TC(Q + D) - TC(Q)
• But D is very small, and eventually approaching 0.
• Problem: when D is approaching 0, both sides of the inequality are 0.
• How do we compare them? We divide both sides by D and let D
approach 0. What we get is the rate of change.
• For perfectly divisible goods, the rule becomes that do it if the rate of
change of the total benefits is larger than the rate of change of the total
costs.
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Understanding Marginalism
in Continuous Case
• We define the marginal benefit and the marginal costs for
perfectly divisible goods as their respective rate of change.
, where D approaches 0.
• Optimal Q is given by MB(Q) = MC(Q)
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Summary:
General Rule for Optimal Quantity
• Discrete Quantity:
• Consume additional units as long as MB(Q) > MC(Q). Stop
otherwise.
• Continuous Quantity:
• Choose Q where MB(Q) = MC(Q).
• A caveat:
• We implicitly assume that MB is non-increasing, and MC is
non-decreasing.
• These assumptions simplify analysis. They need not be true in
reality. We will stick with these assumptions in this course.
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Allocation of Resources
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Recall: The Scarcity Principle
• Key Idea: Resources are limited, but wants are unlimited.
• Implication: Every choice involves trade-offs.
• Examples: time, human resource etc
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Allocation of fixed amount of resources
• When allocating a fixed amount of resources among several
activities, allocate the additional unit of resources to activities with
the highest marginal benefits.
• When the resource is perfectly divisible, allocate the resources
such that MB(A1)=MB(A2)=…=MB(An)
• Assume there does not exist two activities such that one is always better than the other.
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Example – Allocation of Gatherers
• We have 4 gatherers. We can send them out to either east or west side
of a hill to gather mushrooms. We want to maximize our output of
mushrooms.
• Their average output on each side of the hill is as shown below
Average output per gatherer (kg)
Number of gatherers
East
West
1
10
13
2
10
12
3
10
11
4
10
10
• How many gatherers should we send to each side?
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Example – Allocation of Gatherers
• The marginal cost of hiring a gatherer is the same no matter
where we assign her.
• The marginal benefit of assigning her to the East or West differs.
• We should assign each gatherer to the side where her marginal
output is higher.
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Example – Allocation of Gatherers
• First, let’s calculate the MB.
Average output per gatherer (kg)
Number of gatherers
East
West
1
10
13
2
10
12
3
10
11
4
10
10
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Example – Allocation of Gatherers
Average output per gatherer (kg)
Number of gatherers
East
West
1
10
13
2
10
12
3
10
11
4
10
10
Marginal output per gatherer (kg)
Number of gatherers
East
West
1
10
13
2
10
11 (=24-13)
3
10
9 (=33-24)
4
10
7 (=40-33)
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Example – Allocation of Gatherers
Marginal output per gatherer
Number of gatherers
East
West
1
10
13
2
10
11 (=24-13)
3
10
9 (=33-24)
4
10
7 (=40-33)
Should I send the first gatherer to the east?
No. MB(East 1) = 10 < MB(West 1) = 13
Should I send the second gatherer to the east?
No. MB(East 1) = 10 < MB(West 2) = 11
Should I send the third gatherer to the east?
Yes. MB(East 1) = 10 > MB(West 3) = 9
Should I send the fourth gatherer to the east?
Yes. MB(East 2) = 10 > MB(West 3) = 9
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Summary
• Marginal Principle: Compare MB and MC to make decisions.
• Optimal Quantity: Choose Q where MB(Q) = MC(Q).
• Resource Allocation: Allocate resources to maximize marginal
benefits.
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The Invisible hand
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The Rationality Assumption
• The rationality assumption is the foundation of neo-classical
economics.
• The key idea is decision-makers aim to maximize their net benefit.
• Sounds selfish?
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Self-interest vs Social interest
• Ricky Wong, the founder of HKBN, built the fiber-optic network in
HK because his private benefit of doing so is larger than its cost.
• As a result, we have high-speed broadband in Hong Kong.
• This is what Adam Smith called, the invisible hand.
Selfinterest
Market
transactions
Societal
benefits
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“It is not from the
benevolence of the
butcher, the brewer,
or the baker that we
expect our dinner, but
from their regard to
their own interest.”
-Adam Smith,
The Wealth of Nations
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Learning Objectives
At the end of the chapter, students will be able to
• Make predictions and infer information using the cost-benefit
principle
• Understand the assumptions of cost-benefit analysis.
• Apply marginal principle when carrying out cost-benefit analysis.
• Predict the effect of changing incentives.
• Determine opportunity costs in trade-offs.
• Understand the concept of the invisible hand.
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Reminders
• On this Saturday 8am, you may choose a tutorial session on
tutorial sign-up platform.
• On this Saturday 10:30pm the regular quiz for this week will open
on the MOODLE common page. You will have a week of time to
finish it.
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