Nothing Special   »   [go: up one dir, main page]

Topic 1

Download as pdf or txt
Download as pdf or txt
You are on page 1of 42

21/02/2024

Intermediate Microeconomics
Nguyen Thi Hao
haont7@gmail.com
0949.230.527

2
21/02/2024

Contents
Topic 1: Consumer behavior (2 weeks)
Topic 2: Optimal choice (2 weeks)
Topic 3: Consumer choice (2 week) + test 1
Topic 4: Technology, cost and profit (2 weeks)
Topic 5: Pure Competition (2 weeks)
Topic 6: Monopoly (2 weeks) + test 2
Topic 7: Oligopoly (2 weeks)
Topic 8: General Equilibrium (2 weeks) + test 3

4
21/02/2024

Grade
• Attendance, discussion: 10%
Note: absent 1 lesson without permission, -1
score, 2 late lessons = 1 absent day
• Test 1: 20%
There are 3 tests
• Test 2: 10%
choose 2 highest scores

• Final exam: 60%

• Actively involving in lectures earns higher


scores
5

6
21/02/2024

How to study effectively


Printout lecture materials
Before class
Review the previous lessons
Read new lesson
Take note of new vocabulary
Actively involve in lectures
Avoid absent from class without a reason
Straitly ask lecturer if you are confused
Seriously review and sum-up lesson before each test
Making your own mind map of lecture contents
Carefully re-answer question reviews after each chapter
(submit in the end of course to get plus score)
8
21/02/2024

10
21/02/2024

Review
1. Trình bày về đường giới hạn ngân sách (Khái niệm, giải thích, hệ số góc, vẽ đồ thị)
2. Lợi ích (Utility) trong kinh tế là gì? Lợi ích cận biên (Marginal Utility) là gì? Qui luật lợi ích
cận biên giảm dần
3. Đường bàng quang (Indifference curve) là gì? Giải thích bằng đồ thị
4. Tỉ lệ thay thế biên (Marginal Rate of Substitution-MRS) là gì? Công thức, đồ thị minh họa
5. Trình bày sự lựa chọn tối ưu của người tiêu dùng/tối đa hóa độ thỏa dụng? (đồ thị)
6. Một người tiêu dùng có thu nhập là 90.000 đồng và muốn chi vào 2 hàng hóa A và B với
giá hàng hóa A là PA= 30.000 đồng và giá hàng hóa B là PB= 10.000 đồng
- Viết phương trình đường ngân sách và vẽ đồ thị minh họa
- Nếu giá hàng hóa A giảm 50% thì đường ngân sách mới như thế nào?
- Nếu thu nhập của người tiêu dùng tăng gấp đôi thì đường ngân sách mới như thế nào?
11

Contents
Topic 1: Consumer behavior (2 weeks)
Topic 2: Optimal choice (2 weeks)
Topic 3: Consumer choice (2 week) + test 1
Topic 4: Technology, cost and profit (2 weeks)
Topic 5: Pure Competition (2 weeks)
Topic 6: Monopoly (2 weeks) + test 2
Topic 7: Oligopoly (2 weeks)
Topic 8: General Equilibrium (2 weeks) + test 3

12
21/02/2024

Textbook,
Topic 1: Consumer behavior chapter 2,3,4
1.1. Preferences
- Preferences: Assumptions and types (3.1, 3.2)
- Indifference curves (3.3, 3.4, 3.5)
- Marginal rate of substitutions (MRS): Behaviour and interpretation (3.8)
1.2. Budget Constraints
- Budget set (2.1)
- Budget lines and changes (2.3, 2.4, 2.6)
- The numeraire (2.5)
1.3. Utility
- Ordinal utility (4.0)
- Utility functions (4.3)
- Cobb-Douglas preferences (4.3)
1.4. Discussions and practice exercise
- Discussion
- Practice exercises 13

1.1. Preferences
- Preferences: Assumptions and types (3.1, 3.2)
- Indifference curves (3.3, 3.4, 3.5)
- Marginal rate of substitutions (MRS): Behaviour and interpretation (3.8)

14
21/02/2024

1.1. Preferences
• The economic model of consumer behavior: “people choose the best things they can
afford.”
• Call objects of consumer choice: “consumption bundles” (a complete list of the goods and
services).
• Analyzing consumer choice: simplify use two-dimensional diagram consumption bundle X
to consist of two goods:
• x1 denote the amount of one good
• x2 the amount of the other.
•  The complete consumption bundle is therefore denoted by (𝑥 , 𝑥 ).

15

1.1.1. Consumer preferences


• Two consumer bundles: X (𝑥 , 𝑥 ); Y (𝑦 , 𝑦 ). The consumer can rank them
as to their desirability
• Idea of preference: based on consumer’s behaviour: a situation involving 2
bundles, observe consumer’s choice to know which consumer prefer.
• Comparing two different consumption bundles, x and y:
• strict preference: x is more preferred than y: X (𝑥 , 𝑥 ) > Y (𝑦 , 𝑦 )
• weak preference: x is as at least as preferred as y: X (𝑥 , 𝑥 ) ≥ Y (𝑦 , 𝑦 )
• indifference: x is exactly as preferred as y (indifferent): X (𝑥 , 𝑥 ) = Y (𝑦 , 𝑦 )

16
21/02/2024

Assumptions about Preferences


• Three assumption of consumer theory:
1. Completeness : assume any 2 different bundles can be compared.
X (𝑥 , 𝑥 )≥Y (𝑦 , 𝑦 ) or Y (𝑦 , 𝑦 ) ≥ X (𝑥 , 𝑥 ) or both (indifferent)

2. Transitivity: X (𝑥 , 𝑥 )≥Y (𝑦 , 𝑦 ) & Y (𝑦 , 𝑦 ) ≥ Z (𝑧 , 𝑧 )  X (𝑥 , 𝑥 )≥ Z (𝑧 , 𝑧 )

3. Non-satiation: more is better than less


(textbook: Reflexive. We assume that any bundle is at least as good as itself: (x1, x2) (x1, x2) )

17

1.1.2. Indifference Curves (IC)


• Indifferent Curves represents it all
combinations of consumption among
which you are indifferent

• 3 choices:
• A: 2 pizzas; 1 cookie
• B: 1 pizza; 2 cookies
• C: 2 pizza, 2 cookies
• Which choice do you like the most?

18
21/02/2024

x2
x2
WP(x), the set of SP(x), the set of
x bundles weakly x bundles strictly
preferred to x. preferred to x,
does not
WP(x) include
includes I(x).
I(x) I(x). I(x)

x1 x1
Weakly preferred set: The area consist of all bundles that are at least as good as the
bundles (x1, x2)
19

4 properties of IC
1. Consumers prefer higher IC
2. IC are downward slopping
3. IC never cross
4. Only one IC through every bundle

20
21/02/2024

Why does that violate the principle of nonsatiation?

21

• Why this violate the third


property? – IC crossing

B >C
B=A
C=A
 violate transitivity

22
21/02/2024

1.1.3. Marginal rate of substitutions (MRS)

Marginal rate of substitution (MRS): is the slope of the indifference curve at x


MRS measure rate at which the consumer is only just willing to substitute x2 for a small amount of commodity x1.

MRS = Δx2/Δx1 23

1.1.3. Marginal rate of substitutions (MRS)

• The marginal rate of substitution is the slop of an IC at a particular point


• This is a measure of rate at which the consumer is just willing to substitute one
good for the other.
• What we’re doing is taking a little of good one away, Δx1 and replacing it with a
little of good two Δx2, to keep the indifferent.
• MRS = Δx2/Δx1
• Since IC typically have negative slopes, the sign of the MRS is often
negative.

24
21/02/2024

1.1.4. Shapes of indifferent curves


• Well-Behaved Preferences
• Monotonic
• Convex

• Other special shapes of IC:


• Perfect substitutes
• Perfect complements
• Bads
• Neutrals
• Satiation
• Discrete goods

25

Well-Behaved Preferences
Monotonic
• Assume that “more is better”
• This is called “monotonicity” of preferences
• If (x1, x2) is a bundle and (y1, y2) is a bundle
of goods with at least as much of both goods
and more of one, then (y1, y2) > (x1, x2)
• We assume we’re considering points before
satiation occurs
• This assumption implies that indifference
curves will have a negative slope
26
21/02/2024

Well-Behaved Preferences
• Convexity: Mixtures of bundles are (at least weakly) preferred to the
bundles themselves. E.g., the 50-50 mixture of the bundles x and y is
z = (0.5)x + (0.5)y. z is at least as preferred as x or y.
x
x2 x
x2

z =(tx1+(1-t)y1, tx2+(1-t)y2)
x2+y2 x+y
z=
2
2

y2 y y
y2
x1 x1+y1 y1
2 x1 y1
z is preferred to x and y for all 0 < t < 1. 27
z is strictly preferred to both x and y.

x
x2 x’
z’
z

x
z
y y

y2 y’

x1 y1
Preferences are strictly convex Preferences are weakly convex if
when all mixtures z are strictly preferred to their at least one mixture z is equally
component bundles x and y. preferred to a component bundle. 28
21/02/2024

Non-Convex Preferences

x2
x2

z
z

y2 y2

x1 y1 x1 y1

Concave preferences: The mixture z is less Nonconvex preferences: The mixture z is


preferred than x or y. less preferred than x or y.

Why do we want to assume that well-behaved preferences are convex?


29

30
21/02/2024

1.1.4. Some other special shapes of Preferences


• Perfect substitutes
• Perfect complements
• Bads
• Neutrals
• Satiation

31

Perfect substitutes
• Two goods are perfect substitutes if the consumer is willing to substitute one good for the
other at a constant rate.
• The key characteristic is indifferent curves with a constant slope (straight line)

Eg. Considering a choice between red pencils and


Indifference curves are straight
blue pencils, and the consumer involved likes pencils,
lines with a slope of -1
but doesn’t care about color at all
•  Bundles with more total pencils are preferred to
bundles with fewer total pencils
• so the direction of increasing preference is up and
to the right
• The indifference curves for this consumer are all
parallel straight lines with a slope of −1,
indifference curve through (10, 10) or (20, 20) have
a slope of −1
32
21/02/2024

Perfect complements
• Two goods are perfect complements if
consumers are always consumed
together in fixed proportions.
• IC are L-shaped where the vertex occurs
where the number of good 1 equals the
number of good 2.
• Increasing both quantities are the same
time moves the consumer to a higher
indifference curve
• The fixed proportion doesn’t need to be
1-1, could be 2-1, 3-1, …
33

Bad Neutral good


• A commodity is a bad if consumer • A good is a neutral good if consumer
doesn’t like it doesn’t care one way or the other
• IC have a positive slope • If x1 is a normal good, and x2 is neutral
• To put up with good x2 the consumer good then IC are veritcal
need to be given more x1

34
21/02/2024

Satiation
• Satiation occurs where there is an overall best
bundle for the consumer and the “closer” they are
to that best bundle the better off they are terms of
their own preferences
• Suppose (𝑥 , 𝑥 ) is most preferred bundle
• The further from this bundle, the worse off the
consumer is
• Then (𝑥 , 𝑥 ) is called a satiation point or bliss
point

• IC
• Negative slope when consumer has too little or too much of both goods (too much of both
means both are bads)
• Positive slope when consumer has too much of one goods (it becomes a bad, reducing
consumption is better). 35

Summary: Preferences
1. Economists assume that a consumer can rank various consumption
possibilities. The way in which the consumer ranks the consumption
bundles describes the consumer’s preferences.
2. Indifference curves can be used to depict different kinds of
preferences.
3. Well-behaved preferences are monotonic (meaning more is better)
and convex (meaning averages are preferred to extremes).
4. The marginal rate of substitution (MRS) measures the slope of the
indifference curve. This can be interpreted as how much the consumer
is willing to give up of good 2 to acquire more of good 1.
36
21/02/2024

REVIEW QUESTIONS:

Problem 1. Assume you put $1 bill on vertical axis, what is your marginal rate
of substitution of $1 bills for $5 bills?

Problem 2. If we observe a consumer choosing (x1, x2) when (y1, y2) is


available one time, are we justified in concluding that (x1, x2) > (y1, y2)?

Problem 3. If both pepperoni and anchovies are bads, will the indifference
curve have a positive or a negative slope?

37

Problem 4.
• In each of the following examples, the consumer consumes only two
goods, x and y. Based on the information given in each statement,
sketch a plausible set of indifference curves (draw at least two curves
on a set of labeled axes and indicate the direction of higher IC).
• 1. Alan likes wearing both right shoes (x) and left shoes (y). He always
needs to wear them as a pair, having a right shoe is useless without
the left one and vice versa.
• 2. Emma likes pizza (x) but hates vegetables (y). She is only willing to
eat an extra unit of vegetables if she gets to eat an extra unit of pizza.
• 3. Mary likes Coke (x) and Pepsi (y). She is indifferent between them as
she is unable to tell the difference between Coke and Pepsi

38
21/02/2024

Solution 4.

• right shoes and left Bad • Coke and Pepsi are


shoes are perfect perfect substitutes
complements

39

1.2. Budget Constraints

- Budget set (2.1)


- Budget line and changes (2.3, 2.4, 2.6)
- The numeraire (2.5)

40
21/02/2024

1.2.1. Budget set


• The budget set is a set of consumptions which are affordable:

• The set of bundles that exhaust the income is budget line

41

Budget constraint
• Consider a simple 2-good economy
• Suppose a consumer has M to spend on goods 1 and 2
• Consumption bundle (x1, x2) be the amount that consumer chooses of
good 1 and good 2
• p1, p2: Price of two goods
• Consumer can afford all bundles (x1, x2) such: p1x1 + p2x2 = m
•  consumer’s budget set

42
21/02/2024

Budget constraint graphically


x2

vertical interceptm /p2

Not affordable

Budget line: p1x1 + p2x2 = m, slope = -p1/p2


Budget set (affordable)

x1
m /p1
horizontal intercept

Budget line: is a set of bundles that cost exactly m: p1x1 + p2x2 = m


Budget set: Consists all bundles that are affordable at the given prices and income: p1x1 + p2x2 ≤ m
43

Budget line
• Budget set: p1x1 + p2x2 ≤ M, is the set of affordable bundles
• It is bounded by the budget line: p1x1 + p2x2 = M
• This line refers to the bundles that are “just” affordable:
• When the consumer chooses a bundle on the budget line this exhausts the
entire budget
• No money is left over to allocate to one good or the other
• Moving to another bundle requires a tradeoff
• Rearrange the budget line:

44
21/02/2024

Slope of Budget constraint


x2
Slope is -p1/p2

-p1/p2

+1

x1

• Slope is -p1/p2 , How to calculate slope of Budget constraint?


45

Slope of Budget constraint

• Choosing between bundles lying on the budget line: p1x1 + p2x2 = M


require a tradeoff.
• This tradeoff occurs at a rate set by the market: The price ratio, p1/p2
• When consumer wants to stay on the budget line and consumes more
of x1 = ∆𝑥 , he must consume less of x2 = ∆𝑥
• Rewritten constraint line: (𝑥 + ∆𝑥 ) 𝑝 + (𝑥 + ∆𝑥 ) 𝑝 = M

 ∆𝑥 𝑝 + ∆𝑥 𝑝 = 0  =−

• This give the rate at which the market is willing to substitute one good
for the other
46
21/02/2024

1.2.2. Budget line and changes


• When price and incomes change, the set of goods that a consumer can afford changes
as well.
• p1/p2 smaller (p2 increase more than p1)  Budget line flatter
• p1/p2 larger (p1 increase more than p2)  Budget line steeper

x2

Slope Flatter
From -2 to -3 Budget constraint pivots;
slope steeper from -p1/p2 to –p’1/p2

-p’1/p2 -p1/p2
Original budget set
x1

p2 increase p1 increase 47

1.2.2. Budget line and changes


• Income changes (example BC of pizza and cookies, income reduce from 70 to 62)
• Original and new budget constraints are parallel (same slope).

x2 x2

x1 x1

Income decrease  shift inward Income increase  shift outward 48


21/02/2024

1.2.2. Budget line and changes


• Governments often use policies that affect the shape of the budget
line, in particular, taxes, subsidies and rationing:
• How do these tools affect budget line?

49

Taxes
Quantity tax: Value tax (Valorem tax):
• Consumer has to pay a certain • Tax on value (the price) of a good
amount to the Government for instead of quantity of good.
each unit of product • A value tax usually expressed in
purchased percentage term
• Assume that the government • Price change: 𝑝 to (1 + 𝑡)𝑝
imposes a quantity (or unit) tax: Budget line steeper
t usd/unit
• Price change: 𝑝 to (𝑝 +𝑡)
 Budget line steeper

50
21/02/2024

Subsidy is opposite effect of a tax


Quantity Subsidy: Subsidy
• The Government gives an • Base the price of the good being
amount to consumer each unit subsidized (σ)
of good purchased • Actual price: (1- σ)𝑝
• Price change: : 𝑝 to (𝑝 −𝑠) Budget line flatter

 Budget line flatter

51

Lump-sum tax or subsidy


Lump-sum tax: Lump-sum subsidy:
• The Government take away some fixed • The Government give some fixed amount
amount of money, regardless of of money
consumer behavior • Amount of money has been increased
• Amount of money has been reduced •  Budget line shift outward
•  Budget line shift inward

52
21/02/2024

Rationing

• Rationing: The level of consumption of


some product is fixed to be no larger
than some amount:
• Suppose good 1 were rationed 𝑥̅ < 𝑥
• Budget set with a piece lopped off affordable but have
𝑥 > 𝑥̅

53

Taxes, subsidies, and rationing are combined


• Consumer could consume good 1
at a price of 𝑝 up to level of 𝑥̅
• They have to pay a tax t on all
consumption in excess of 𝑥̅

• If:
𝑥 < 𝑥̅  Slope = - 𝑝 /𝑝
 𝑥̅ < 𝑥  Slope = - (𝑝 +𝑡)/𝑝 ,
budget line steeper

54
21/02/2024

Example: Food stamp program

• Read textbook (page 29-31) 55

1.2.3. The numeraire


• The budget line is defined by two prices and one income, but one of these
variables is redundant. We could peg one of the prices, or the income, to
some fixed value, and adjust the other variables so as to describe exactly the
same budget set
p1x1 + p2x2 = m
• We can transform the budget equation into other type
or

56
21/02/2024

• In the first case, we have pegged p2 = 1, and in the second case m = 1.


Pegging the price of one of the goods or income to 1 and adjusting the
other price and income appropriately doesn’t change the budget set.
• When we set one of the prices to 1, we refer to that price as the
numeraire price. The numeraire price is the price relative to which we
are measuring the other price and income.

57

REVIEW QUESTIONS
1. Originally the consumer faces the budget line . Then the
price of good 1 doubles, the price of good 2 becomes 8 times larger, and income
becomes 4 times larger. Write down an equation for the new budget line in terms
of the original prices and income.
2. What happens to the budget line if the price of good 2 increases, but the price
of good 1 and income remain constant?
3. If the price of good 1 doubles and the price of good 2 triples, does the budget
line become flatter or steeper?
4. Suppose that a budget equation is given by . The
government decides to impose a lump-sum tax of u, a quantity tax on good 1 of
t, and a quantity subsidy on good 2 of s. What is the formula for the new budget
line?

58
21/02/2024

• Problem 1: You have an income of $40 to spend on two commodities. Commodity 1 costs $10 per
unit, and commodity 2 costs $5 per unit.
(a) Write down your budget equation. .
(b) If you spent all your income on commodity 1, how much could you buy? .
(c) If you spent all of your income on commodity 2, how much could you buy?
(d) Suppose that the price of commodity 1 falls to $5 while everything else stays the same. Write
down your new budget equation
(e) Suppose that the amount you are allowed to spend falls to $30, while the prices of both
commodities remain at $5. Write down your budget equation.
(f ) Draw your budget line changes, show area representing commodity bundles that you can afford
with the budget in Part (e) but could not afford to buy with the budget in Part (a). And the area
representing commodity bundles that you could afford with the budget in Part (a) but cannot afford
with the budget in Part (e).
• Problem 2. Murphy was consuming 100 units of X and 50 units of Y . The price of X rose from 2 to
3. The price of Y remained at 4.
How much would Murphy’s income have to rise so that he can still exactly afford 100 units of X
and 50 units of Y ?
• Problem 3. If Amy spent her entire allowance, she could afford 8 candy bars and 8 comic books a
week. She could also just afford 10 candy bars and 4 comic books a week. The price of a candy bar
is 50 cents. Draw her budget line, what is Amy’s weekly allowance? 59

Summary: Budget constraint


1. The budget set consists of all bundles of goods that the consumer can
afford at given prices and income. We will typically assume that there are
only two goods, but this assumption is more general than it seems.
2. The budget line is wri en as p1x1 +p2x2 = m. It has a slope of −p1/p2, a
vertical intercept of m/p2, and a horizontal intercept of m/p1.
3. Increasing income shifts the budget line outward. Increasing the price of
good 1 makes the budget line steeper. Increasing the price of good 2 makes
the budget line flatter.
4. Taxes, subsidies, and rationing change the slope and position of the
budget line by changing the prices paid by the consumer.

60
21/02/2024

1.3. Utility

61

1.3. Utility Text book chapter 4

Point of views:
- Historically, Utility was thought as a numeric measure of a consumer’s happiness 
Consumer try to maximize utility
- Questions: How do we measure?
How do we quantify utility from different choices
How do we compare utility between people
- Instead of above, start to think about utility as being constructive from preferences,
preferences describe choices
Utility is seen only as a way to describe preferences

62
21/02/2024

1.3.1. Utility function and ordinal utility


• Utility function: Takes consumption bundles and translates them into number
• More preferred bundles have higher utility than less preferred bundles:
(x1, x2) > (y1, y2) if and only if u(x1, x2) > u (y1, y2)
• Denote: u(.): utility function

• Ordinal utility: Absolute numbers do not matter, just the ranking of the
bundles
Size of utility difference between bundles does not matter
This kind of utility is referred to as ordinal utility.

63

Examples
• order the bundles in the same way
• consumer prefers A to B and B to C
• All of the ways indicated are valid utility functions that describe the same
preferences

64
21/02/2024

1.3.2. Monotonic transformations


• Since only the ranking matter, no unique way to assign utility to
consumption bundles  infinite number of way
• Ex: if u(x1, x2) present a way to assign utility numbers to the bundles (x1,
x2) then multiplying u(x1, x2) by 2 (or any positive number), that is an
other way to assign utility.
• (positive) Monotonic function: Transform a set of number into another
set that maintains the order of the numbers. If u1 > u2  f (u1) > f (u2)
• present a monotonic transformation by a function f(u)

65

• Examples: Multiply by a positive number (f(u) = 3u), adding any number (f(u) = u
+ 7), raising u to an odd power (f(u) = 𝑢 ), ….
• The rate of change of f(u) as u changes

• Always has a positive rate of change  This means that the graph of a
monotonic function will always have a positive slope

Not monotonic function, sometimes


Monotonic function, always increasing
increases, sometimes decreases.
66
21/02/2024

Which of the following are monotonic transformations?


(1) u = 2v − 13;
(2) u = −1/𝑣 ;
The text said that raising a number to an odd
(3) u = 1/𝑣 ; power was a monotonic transformation. What
(4) u = ln v; about raising a number to an even power? Is
this a monotonic transformation?
(5) u = −𝑒 ; (Hint: consider the case f(u) = u2.)
(6) u = 𝑣 ;
(7) u = 𝑣 for v > 0;
(8) u = 𝑣 for v < 0

67

Principle: a monotonic transformation of a utility function is a utility


function that represents the same preferences as the original utility
function.

• If f(u) is any monotonic transformation of a utility function that


represents some particular preferences, then f(u(x1, x2)) is also a
utility function that represents those same preferences.
1. To say that u(x1, x2) represents some particular preferences means
that u(x1, x2) > u(y1, y2) if and only if (x1, x2) > (y1, y2).
2. But if f(u) is a monotonic transformation, then u(x1, x2) > u(y1, y2) if
and only if f(u(x1, x2)) > f(u(y1, y2)).
3. Therefore, f(u(x1, x2)) > f(u(y1, y2)) if and only if (x1, x2) > (y1, y2), so
the function f(u) represents the preferences in the same way as the
original utility function u(x1, x2).
68
21/02/2024

1.3.3. Constructing a utility function from indifference curves


• rule out cases of intransitive preferences
• Suppose that we are given an indifference
map
• If preferences are monotonic then the line
through the origin must intersect every
indifference curve exactly once.
• Thus every bundle is getting a label, and
those bundles on higher indifference curves
are getting larger labels
• All bundles in an indifference curve have the
same utility level.
• The collection of all ICs for a given preference
relation is an indifference map.
• An indifference map is equivalent to a utility function. Draw a diagonal line and label each indifference
curve with how far it is from the origin measured
69
along the line

Some examples of utility functions


• represent these preferences by utility functions.

• Function: u(x1, x2)


• Draw IC: Plot all point (x1,x2) that have u(x1,x2) = k
• For each difference value of k, get a different IC, for
example, k = 1 we have the lowest IC, and with k = 3,
we have highest IC
• Level set: Set of all (x1,x2) such that u(x1, x2) = a
constant (k).

70
21/02/2024

Eg1: Suppose that the utility function is given by: u(x1, x2) = x1x2.
What do the indifference curves look like?

• We know that a typical indifference


curve is just the set of all x1 and x2
such that k = x1x2 for some constant k.
• Solving for x2 as a function of x1, we
see that a typical indifference curve
has the formula: x2 = k/x1
• This curve is depicted in Figure for k =
1, 2, 3 · · ·.

71

Eg2: Utility function:


How do its indifference curves look like?

Since u(x1, x2) > 0  v(x1,x2) is monotonic transformation of u(x1,x2)


v(x1,x2) has exactly the same shaped indifferent curves of u(x1, x2) (previous slide)

 The labeling of IC will be different


The label 1, 2, 3, … now will be 1, 4, 9, …
The set bundles that v(x1, x2) = 9, exactly the same
as the set of bundles that has u(x1, x2) =3

72
21/02/2024

Perfect substitution
• The red pencil and blue pencil example: All the matter to the consumer was the total
number of pencils.
• Measure utility by the total number of pencils: u(x1, x2) = x1+x2
• Does this work? Just ask two things: is this utility function constant along the indifference curves? Does it
assign a higher label to more-preferred bundles? The answer to both questions is yes,  a utility function.

• if substitute rate good 1 and good 2 = 1:1  Utility function u(x1, x2) = x1+x2
• Any monotonic transformation of u(x1, x2) will also represent the perfect substitutes preferences, ,
ex: v(x1, x2) = 𝑥 + 𝑥 = 𝑥 + 2𝑥 𝑥 + 𝑥
• if substitute rate good 1 and good 2 ≠ 1:1 (VALUE 2𝑥 substitutes for 1𝑥 ) we have an
utility function: u(x1,x2) = 2𝑥 + 𝑥  Slope = -2

 General utility function present perfect substitution preferences:


u(x1,x2) = 𝐚𝒙𝟏 + 𝒃𝒙𝟐 , slope = -a/b
1 73

Perfect complement
x2

45o
• Remind example of left shoe and right shoe
• Consumer care about the number of pairs of
shoe  Choose the number of pairs of
shoes as the utility function min{x1,x2} = 8
8
• The number of complete pairs: minimum of
the number right shoes and the number of 5 min{x1,x2} = 5
left shoes, thus the utility function for
perfect complement: u(x1,x2) = min{x1,x2} 3 min{x1,x2} = 3

• Ex: take bundle min{10,10} = min{11,10} = x1


10 3 5 8

74
21/02/2024

• Second example: consumer always uses 2 teaspoons of sugar with


each cup of tea
• If x1 is the number of cups of tea available and x2 is the number of
teaspoons of sugar, utility function: u(x1,x2) = min{𝑥 𝑥 }.
• In general, a utility function that describes perfect-complement
preferences is given by:
u(x1, x2) = min{a𝑥 + 𝑏𝑥 }
• a and b are positive numbers that indicate the proportions in which the
goods are consumed

75

Quasi-linear utility function


• The height of each difference curve is some
function of x1 plus a constant. There’s a general x2
form: U(x1,x2) = k = v(x1) + x2 Each curve is a vertically
shifted copy of the others.
• Quasi-linear (partly linear): The utility function is
linear in good 2, but (maybe) nonlinear in good 1
• E.g.:
• u(x1,x2) = ln x1 + x2;
• u(x1,x2) = 𝑥 + 𝑥 ;
• u(x1,x2) = 𝑥 + 𝑥
• Quasi-linear utility function : the preferences are
vertically shifted version:
x1
x2= k – v(x1)
76
21/02/2024

Cobb-Douglas utility function


• Cobb-Douglas utility function:
u(x1,x2) = x1a x2b
with a > 0 and b > 0.
• E.g. U(x1,x2) = x11/2 x21/2 (a = b = 1/2)
V(x1,x2) = x1 x23 (a = 1, b = 3)
• Cobb-Douglas preferences are the
standard example of IC that look well-
behaved
• All curves are hyperbolic,
asymptoting to, but never
touching any axis.
77

1.3.4. Marginal Utilities (MU)


• The marginal utility: measures the rate of change in utility (U) associated
with a small change in the amount of good 1(x1) or good 2 (x2)

• MU if x1 change:

• MU if x2 change:

78
21/02/2024

1.3.5. Marginal Utility and Marginal Rate of


Substitution (MU and MRS)
• Consider a change in the consumption of each good, (Δx1, Δx2), that keeps
utility constant—U(x1,x2)  k (k is a constant), that is, a change in consumption
that moves us along the indifference curve. Then we must have:
• Totally differentiating this identity gives: 𝑀𝑈 𝑥 + 𝑀𝑈 𝑥 = 𝑈 = 0

 MRS = =-

• Negative sign: If get more of good 1, have to get less of good 2 in order to keep
the same level of utility.
• However, MRS is normally refer by its absolute value – as a positive number

79

Monotonic Transformations & MRS


• Eg: U(x1,x2) = x1x2, the MRS = -x2/x1.
• Create V = U2; i.e. V(x1,x2) = x12x22.
• What is the MRS for V?
 V /  x1 2 x1x22 x
MRS     2
 V /  x2 2 x12 x2 x1

• which is the same as the MRS for U.

80
21/02/2024

Summary
1. A utility function is simply a way to represent or summarize
a preference ordering. The numerical magnitudes of utility
levels have no intrinsic meaning.
2. Thus, given any one utility function, any monotonic
transformation of it will represent the same preferences.
3. The marginal rate of substitution, MRS, can be calculated
from the utility function via the formula MRS = Δx2/Δx1 =
−MU1/MU2.

81

Review questions

82
21/02/2024

83

You might also like