Economics Lecture 2 - Univerzita Karlovavitali/documentsCourses/Lecture 2 - Prefere… · 1 u(x 1,...
Transcript of Economics Lecture 2 - Univerzita Karlovavitali/documentsCourses/Lecture 2 - Prefere… · 1 u(x 1,...
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Economics Lecture 2
2016-17
Sebastiano Vitali
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Course Outline
1 Consumer theory and its applications
1.1 Preferences and utility
1.2 Utility maximization and uncompensated demand
1.3 Expenditure minimization and compensated demand
1.4 Price changes and welfare
1.5 Labour supply, taxes and benefits
1.6 Saving and borrowing
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2 Firms, costs and profit maximization
2.1 Firms and costs
2.2 Profit maximization and costs for a price taking firm
3. Industrial organization
3.1 Perfect competition and monopoly
3.2 Oligopoly and games
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1.1 Preferences and utility
List of topics
1. Marshall’s model of consumer demand
2. Hicks’ model of preferences and utility
3. Modern assumptions on preferences and utility:
1. Completeness
2. Transitivity
3. Continuity
4. Non-satiation
5. Convexity
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Preferences and utility
2. Marshall’s model of
consumer demand
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Consumer theory and its applications
• Late 19th century orthodoxy Marshall.
• Principles of Economics, 1890
• Mid 20th century orthodoxy Hicks,
• Value & Capital (1939)
builds on Edgeworth (1881), Pareto (1909), Slutsky (1915)
• The modern textbook theory .
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The purpose of consumer theory
• to understand how household behaviour
– buying, selling, working, saving and borrowing
• is affected by changes in
– prices, income, wages, interest rates, tax rates …..
• to provide a way of
– showing that there are situations in which everyone gains from trade
– measuring the effect on household welfare of changes in prices, income etc..
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• Cambridge
• Principles of Economics (1890)
• supply and demand curves
• elasticity
• consumer surplus
Alfred Marshall (1842-1924)
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Marshall’s model of consumer demand
• roots in 19th century utilitarian philosophy
• achievements
– the demand curve
– consumer surplus
• gaps
– lots of words, no modelling of
the effects of income & prices of other goods
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Marshall’s theory of
consumer demand with
integer quantities,
1, 2, 3…
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Marshall’s assumptions on consumer
behaviour
• The consumer chooses quantity x to maximize V(x) – px.
• The price p does not depend on x
• V(x) is measured in units of money, €, V(0) = 0.
• V(x) is the most a consumer would be willing to pay for x units
if the alternative were 0 units.
• Marshall calls V(x) “utility”
• I call V(x) value to avoid confusion with the modern
interpretation of utility.
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• Marginal value MV(x) = V(x) – V(x-1).
maximum willing to pay for one more unit.
• Marginal value is positive.
• Marginal value decreases as x increases.
Marshall’s assumptions on consumer
behaviour
Marshall notes marginal utility may increase with
consumption for some goods – e.g. for good
music.
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x
total value V(x)
increases but
at a
decreasing
rate
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x
MV(3) = V(3) – V(2) marginal value
marginal value
= gradient of “tangent line”
marginal value is positive,
but decreases as x
increases.
2 3
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Marginal conditions with integers
• p = price of x
• Objective is to max V(x) – px
• MV(x) = V(x ) – V(x - 1 ) > 0 but decreases as x increases.
• If MV(x) > p increasing from x -1 to x increases V(x) – px
• If MV(x + 1) < p increasing from x to x + 1 decreases V(x) – px
• If MV(x) is decreasing then x is optimal if
MV(x + 1 ) ≤ p ≤ MV(x)
• If MV(1) < p then x = 0 is optimal.
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0
100
200
300
400
500
600
1 2 3 4 5 6 7 8 9 10 11 12 13 14 x
MV(x)
p
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0
100
200
300
400
500
600
1 2 3 4 5 6 7 8 9 10 11 12 13 14 x
MV(x)
p MV(1) = 500
so x = 0 is optimal if p > 500
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0
100
200
300
400
500
600
1 2 3 4 5 6 7 8 9 10 11 12 13 14 x
MV(x)
p MV(10) < 200 < MV(9)
so x = 9 is optimal if p = 200
marginal value “curve” is the
demand curve.
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V(0) = 0
MV(1) = V(1) - V(0) = V(1)
MV(2) = V(2) – V(1)
MV(3) = V(3) – V(2)
Add these equations
MV(1) + MV(2) + MV(3) = V(3)
Marginal value MV and total value V
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x
marginal value
Total value = V(x)
= sum marginal values
= area under marginal value “curve”
total value
Consumer surplus
in Marshall’s model
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x
marginal value
total cost = px
Consumer surplus
in Marshall’s model
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x
marginal value
Consumer surplus
= total value – total cost
= gain from trade.
Consumer surplus
in Marshall’s model
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Marshall’s theory with
calculus
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Move to calculus
• Done by working with smaller and smaller units of
output.
• Comes from taking limits as the unit size tends to 0.
• Marginal value = gradient of tangent to total curve
• Differentiate total value
• Total value = area under marginal value curve.
• Integrate marginal value.
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• Total value is an increasing function of x so
marginal value > 0
• Marginal value decreasing
• Consumers maximize total value – total cost = V(x) - px
dx
dV
02
2
dx
Vd
Marshall’s assumptions on consumer
behaviour in calculus
dx
dV
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= marginal value > 0
tangent slopes upwards
total value V(x) is
increasing
0 x
V(x)
dx
dV
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0 x
V(x)
02
2
dx
Vd
dx
dV= marginal value is
decreasing
V(x) is concave
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0 x
f(x)Facts about
concave
functions
A function is concave if any line joining two
points on its graph lies entirely on or below the
graph.
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0 x
f(x)Facts about
concave
functions
For functions with first and second derivatives
Concave functions are functions with
decreasing first derivatives
Concave functions are functions with negative
second derivatives.
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0 x0 x
f(x)Facts about
concave
functions
If a function is concave and its derivative is 0 at
x0 then x0 maximizes the function.
Next slides shows why this is implied by
decreasing.
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0 x0 x
Facts about
concave
functions
> 0
f(x)
increasing
< 0
f(x)
decreasing
At x0 = 0 thus x0 maximizes f(x).
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Total value is a concave function of x implies that
total value – total cost = V(x) – px is concave.
Why?
As V(x) is concave the second derivative
The second derivative of V(x) – px is also
so V(x) – px is concave.
02
2
dx
Vd
02
2
dx
Vd
Marshall’s theory with calculus and
concavity
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As V(x) – px is if x0 > 0 concave and the first order
condition
= p
marginal value = price
is satisfied then x0 maximizes total value – total cost
dx
xdV )( 0
Marshall’s theory with calculus and
concavity
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0 xA x
p
pA
Marginal value
= demand
Why if MV= pA at x = xA then xA maximizes V(x) – pAx?
If x < xA then MV > pA, increasing x increases V(x) – px
If x > xA then MV < pA, increasing x decreases V(x) - px
Marshall’s Theory with Calculus
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0 xA x
p
pA
Marshall’s Theory with Calculus
marginal value = demand
As MV is downward sloping if p > MV(0)
then p > MV(x) for all x
optimal to buy 0 units.
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0 xA x
p
pA
Marshall’s Theory with Calculus
Marginal value
= demand
total
value
total value
= area under demand curve
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0 xA x
p
pA
Marshall’s Theory with Calculus
Marginal value
= demandtotal
cost
consumer
surplus
Consumer surplus
= total value - total cost
= gain from trade.
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0 xA x
p
pA
Marshall’s Theory with Calculus
Marginal value
= demand
consumer
surplus
Measuring consumer
surplus is difficult
It requires knowing the
whole demand curve.
total
cost
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0 xA x
p
pB
pA
Marshall’s Theory with Calculus
total cost
Measuring the fall in
consumer surplus
when p rises from pA
to pB requires knowing
demand at pA and pB
and the elasticity of
demand.
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0 x
p
High consumer
surplus low price,
e.g. water
0 x
p
low consumer
surplus high price,
e.g. diamonds
Diamond water paradox
Water is essential, diamonds are not
But diamonds have a much higher price
than water.
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Water, high total value,
low price, low marginal value,
large Consumer Surplus.
Marshall’s theory explains why an
essential good (water) has a much lower
price than a luxury (diamonds).
Diamonds, low total value,
high price, high marginal value,
small Consumer Surplus.
0 x
p
0 x
p
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What has Marshall achieved?
1. Explained the downward sloping demand curve.
2. Demonstrated that consumers gain from trade.
3. Giving a monetary measure of the gain from trade:
consumer surplus – but this requires knowing the whole
demand curve.
4. Explained why a good may have a low price but high
total value, price = marginal value, not total value.
Diamond water paradox.
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What are the difficulties with Marshall’s
model ?
• It does not distinguish between
• changes in demand due to changes in the price of
other goods & income
• changes in demand due to changes in tastes, e.g.
fashion.
• What is utility and can it be measured in money?
• This is a big 19th century philosophical debate.
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Preferences and utility
2. Hicks’ model of preferences
and utility
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General utility functions
Marshall has no model of how the utilities of two goods interact.
Move to utility depending on quantities of all goods,
Consuming x1, x2 ….xn of goods 1,2…n gives utility
u(x1, x2 ….xn) which is a number. This term n = 2.
Note the implicit assumption, utility depends on the goods you
consume, can be extended to other people’s consumption
There is no mention of human relationships in this theory.
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Implications of the utility function
1: completeness
Because utility is a number for any two bundles of goods (x1A,x2A)
and (x1B,x2B) one of these relationships must hold:
u(x1A,x2A) > u(x1B,x2B) the consumer prefers (x1A,x2A) to (x1B,x2B)
u(x1A,x2A) = u(x1B,x2B) the consumer is indifferent between(x1A,x2A)
& (x1B,x2B).
u(x1A,x2A) < u(x1B,x2B) the consumer prefers (x1B,x2B) to (x1A,x2A)
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Implications of the utility function
2: transitivity
Because utility is a number for any three bundles of goods (x1A,x2A)
and (x1B,x2B) and (x1C,x2C)
If u(x1A,x2A) > u(x1B,x2B) and u(x1B,x2B) > u(x1C,x2C)
then u(x1A,x2A) > u(x1C,x2C).
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x2
0 x1
u(x1, x2) = 3
u(x1, x2) = 2
u(x1, x2) = 1
All the points on an indifference
curve have the same utility
so consumers are indifferent
between two points on the same
indifference curve.
There is an indifference curve
through every point but we only
show some of them.
Without further assumptions
indifference curves can have a
strange shape.
Implications of the utility function
3: indifference curves
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John Hicks,
1904 – 89
LSE 1926 to 1935
Nobel prize 1972
Source picture LSE website
Hicks: “Value & Capital”
2nd edition, OUP 1939
Building on
Pareto (1909)
Edgeworth (1891)
Slutsky (1915)
Hicks worked with indifference curves
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3. Hicks on indifference curves: Pareto’s theory
“In order to determine the quantities of goods which an
individual will buy at give prices,
Marshall’s theory implies that we must know his utility
surface; (i.e. utility function);
Pareto’s theory only assumes that we must know his
indifference map.
And that conveys less information than the utility surface.
It only tells us that that the individual prefers one particular
collection of goods to another …..; it does not tell us by
how much the first collection is preferred to the second.”
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Indifference curves
x2
0 x1
u(x1, x2) = 3
u(x1, x2) = 2
u(x1, x2) = 1
Does replacing 1, 2, 3 by 1, 4,
9 change preferences?
Does replacing 1, 2, 3 by
1, 30, 14 change preferences?
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Indifference curves
x2
0 x1
u(x1, x2) = 3
u(x1, x2) = 2
u(x1, x2) = 1
Does replacing 1, 2, 3 by
10,100,1000, change
preferences? No
Does replacing 1, 2, 3 by
1, 30, 14 change preferences?
Yes
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Hicks made two assumptions on the shape
of indifference curve
• Nonsatiation, more is better.
• Decreasing marginal rate of substitution
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Nonsatiation: more is better
x2
0 x1
Consumers prefer more to less.
B is preferred to A because
there is more good 1 and the
same amount of good 2.
To get from B to C on the same
indifference curve good 2 must
be reduced.
A B
C
Nonsatiation implies downward sloping
indifference curves.
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Hicks introduced the term “Marginal Rate of
Substitution”, MRS
x2
0 x1
A B
C
BC
AB= - slope of AC
= average amount of
good 2 needed to
compensate for loosing
1 unit of good 1.
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x2
0 x1
A
Hicks assumed decreasing MRS
MRS = - gradient of indifference
curve
Hicks assumes that the MRS
decreases along an indifference
curve as x1 increases.
With more x1 the consumer
requires less extra x2 to make up
for loosing a unit of x1.
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Summary
• Marshall started his analysis with a utility functions.
• Hicks dropped utility functions and started his analysis
with indifference curves.
• The modern theory starts with preferences
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Preferences and utility
3. Modern assumptions on
preferences and utility
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Modern assumptions on preferences
1. Completeness
2. Transitivity
3. Continuity
4. Nonsatiation
5. Convexity
For each of these
assumption I give
•a definition
•an intuitive explanation
•a critical discussion.
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Modern assumptions on
preferences:
completeness
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Modern Assumption 1: Completeness
If a consumer is choosing between two
bundles A and B one of the following
possibilities holds
she prefers A to B
she prefers B to A
she is indifferent between A and B.
x2
0 x1
. A
. B
This term we work with 2 goods so consumption bundles
can be shown in a diagram.
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Choices over Chinese vegetables
jay-lan
si-gua
mao-gua
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Consumers may find it impossible to rank some options, or do so in a way that is in some sense wrong for them.
Decision making takes time and effort, we are often
ignorant
uncertain
unable to understand what a product is and does
subject to systematic biases
This standard model is not based on psychological research.
Behavioural economics is based on psychological research.
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Modern assumptions on
preferences:
transitivity
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Modern Assumption 2: Transitivity
Transitivity if A is preferred to B and B to C then A is
preferred to C.
Is the relationship “A is taller than B” is transitive?
Is the relationship “team A beat team B last time they
played” transitive?
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Modern Assumption 2: Transitivity
Transitivity if A is preferred to B and B to C then A is
preferred to C.
Is the relationship “A is taller than B” is transitive?
Is the relationship “team A beat team B last time they
played” transitive?
yes
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Modern Assumption 2: Transitivity
Transitivity if A is preferred to B and B to C then A is
preferred to C.
Is the relationship “A is taller than B” is transitive?
Is the relationship “team A beat team B last time they
played” transitive?
yes
no
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Premier League 2015/16
Leicester beat Chelsea
Chelsea beat Arsenal
Arsenal beat Leicester
Transitivity would imply that Leicester beat Arsenal.
The relationship beat in sport is not transitive.
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Premier League 2015/16
Leicester beat Chelsea
Chelsea beat Arsenal
Arsenal beat Leicester
Transitivity would imply that Leicester beat Arsenal.
The relationship beat in sport is not transitive.
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Modern assumptions on
preferences:
continuity
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Modern Assumption 3: Continuity
x1
x2
. A. B
. C
This can be stated much more precisely but not in this
course.
If A is preferred to B
& B is very close to C
then A is preferred to C.
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Summary: Modern theory starts with a
preference relation and assumes
1. Completeness, if A and B are two bundles of goods
then
• either A is preferred to B,
• or A and B are indifferent
• or B is preferred to A
2. Transitivity if A is preferred to B and B to C then A is
preferred to C
3. Continuity if A is preferred to B and C is close to B then
A is preferred to C.
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The completeness, transitivity and continuity
assumptions imply
• preferences can be represented by a utility function and
indifference curves.
• Take this on trust.
• Utility is ordinal i.e. different utility functions can
represent the same preferences.
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Consumer theory:
ordinal utility
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Indifference curves
x2
0 x1
u(x1, x2) = 3
u(x1, x2) = 2
u(x1, x2) = 1
Replacing 1, 2, 3 by 1, 4, 9,
does not change preferences.
Replacing 1, 2, 3 by
1, 30, 14 does change
preferences
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Different utility functions representing the
same preferences
• Suppose utility u(x1,x2) = x13/4 x2
1/2
• If x1 > 0 and x2 > 0 u > 0
• Then as u2 is an increasing function of u (for u>0)
if v = u2 then
v(x1,x2) =(u(x1,x2) )2 = x1
3/2 x2 represents the same
preferences as u(x1,x2)
• This is because the order of numbers attached to
indifference curves does not change.
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Different utility functions representing the
same preferences
• Suppose utility u(x1,x2) = x13/4 x2
1/2
• If x1 > 0 and x2 > 0 u > 0
• Then as ln u is an increasing function of u (for u>0)
if w = ln u then
w(x1,x2) = ln(u(x1,x2) ) = (3/4) ln x1 + (1/2) ln x2
represents the same preferences as u(x1,x2)
• This is because the order of numbers attached to
indifference curves does not change.
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Different utility functions representing the
same preferences
• Suppose utility u(x1,x2) = min [2x1, 3x2]
• If x1 > 0 and x2 > 0 u > 0
• Then as 2u is an increasing function of u
if w = 2 u then
z(x1,x2) = 2 min [2x1, 3x2] = min[4x1, 6x2]
represents the same preferences as u(x1,x2)
• This is because the order of numbers attached to
indifference curves does not change.
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Different utility functions representing the
same preferences
• If f(u) is an increasing function of u
• Then f(u(x)) represents the same preferences as u(x).
• Utility is ordinal not cardinal.
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Modern assumptions on
preferences:
nonsatiation
More is better.
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Nonsatiation in Nicholson, Snyder, Stewart
Section on economic goods
“… the variables are taken to be “goods”, i.e. whatever
quantities they represent we assume that more of any
particular good …. is preferred to less.”
A consumer prefers having more of either good to having
less.
If x1B > x1A then (x1B,x2A) is preferred to (x1A,x2A)
If x2B > x2A then (x1A,x2B) is preferred to (x1A,x2A)
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Nonsatiation in the indifference curve
diagram
x2
0 x1
Nonsatiation means that any
point such as C inside or on the
boundary of the shaded area is
preferred to A.
Here starting from A increasing
x1 and/or increasing x2
increases utility.
. C
A
Nonsatiation implies indifference curves slope
downwards
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0
0
52
2
52
153
53
2
53
152
2
1
//
xu
//
xu
xx
,xx
0021
xu
xu ,If
then the nonsatiation assumption is satisfied
e.g. if u = x12/5x2
3/5
so the nonsatiation assumption is satisfied.
5. Checking the nonsatiation assumption
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If
then increasing x1 or x2 increases utility.
The indifference curve slopes downwards.
The preferred set is above the indifference curve.
Nonsatiation is satisfied.
5. Checking the nonsatiation assumption
0021
xu
xu ,
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Nonsatiation in the indifference curve
diagram
Nonsatiation implies that points
above an indifference curve are
preferred to points on the
indifference curve.
These points are in the
preferred set.
Nonsatiation implies downward sloping
indifference curves.
x2
0 x1
preferred
set
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x2
0 x1
preferred set
Is nonsatiation satisfied here?
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x2
0 x1
preferred set
Is nonsatiation satisfied here?
NO
Drawing indifference
curves like this is an
easy mistake.
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Perfect complements utility: requires a different
treatment of nonsatiation.
0 wheels x1
frames
x2
x2 = ½ x1
u(x1,x2) = min( ½ x1,x2)
x1 bicycle wheels,
x2 bicycle frames
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Modern assumptions on
preferences:
convexity
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Mathematically a set is convex if any straight line joining
two points in the set lies in the set.
Which of these sets are convex?
A B
C D
Modern Assumption 5: Convexity
The preferred set is convex.
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Mathematically a set is convex if any straight line joining
two points in the set lies in the set.
Which of these sets are convex?
A B
C D
Modern Assumption 5: Convexity
The preferred set is convex.
not convex
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Mathematically a set is convex if any straight line joining
two points in the set lies in the set.
Which of these sets are convex?
A B
C D
Modern Assumption 5: Convexity
The preferred set is convex.
convexnot convex
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Mathematically a set is convex if any straight line joining
two points in the set lies in the set.
Which of these sets are convex?
A B
C D
Modern Assumption 5: Convexity
The preferred set is convex.
convex
convexnot convex
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Mathematically a set is convex if any straight line joining
two points in the set lies in the set.
Which of these sets are convex?
A B
C D
Modern Assumption 5: Convexity
The preferred set is convex.
convex
convexnot convex
not convex
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x2
0 x1
preferred
set
Modern Assumption 5: Convexity
The preferred set is convex.
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Interpreting the convexity assumption
x1
B is the of A and C
Convexity implies that B is preferred to
Convexity means preference
A and C are indifferent.
x1
x2
0
preferred
setB
A
C
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Interpreting the convexity assumption
B is the average of A and C
Convexity implies that B is preferred to
Convexity means preference
A and C are indifferent.x2
preferred
setB
A
C
0
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Interpreting the convexity assumption
B is the average of A and C
Convexity implies that B is preferred to A and C.
Convexity means preference
A and C are indifferent.x2
preferred
setB
A
C
0
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Interpreting the convexity assumption
B is the average of A and C
Convexity implies that B is preferred to A and C.
Convexity means preference for averages.
A and C are indifferent.x2
preferred
setB
A
C
0
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Nonconvex preferences with nonsatiation
x2preferred
set
possible examples of nonconvex preferences
tea & coffee consumed simultaneously
alcohol & drugs© Getty Images
0 x1
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x2
0 x1
preferred
set
set
preferred
set
x2
Figure 1 Figure 2 Figure 3
x2 preferred
set
convexity
nonsatiation
convexity
nonsatiation
convexity
nonsatiation
Do these preferences satisfy convexity
and nonsatiation?
TP
0 x10 x1
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x2 preferred
set
set
preferred
set
x2
Figure 1 Figure 2 Figure 3
x2 preferred
set
convexity no
nonsatiation yes
convexity
nonsatiation
convexity
nonsatiation
Do these preferences satisfy convexity
and nonsatiation?
0 x1 0 x10 x1
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x2 preferred
set
set
preferred
set
x2
Figure 1 Figure 2 Figure 3
x2 preferred
set
convexity no
nonsatiation yes
convexity yes
nonsatiation no
convexity
nonsatiation
Do these preferences satisfy convexity
and nonsatiation?
0 x1 0 x10 x1
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x2 preferred
set
set
preferred
set
x2
Figure 1 Figure 2 Figure 3
x2 preferred
set
convexity no
nonsatiation yes
convexity yes
nonsatiation no
convexity no
nonsatiation no
Do these preferences satisfy convexity
and nonsatiation?
0 x1 0 x10 x1
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x2
0 x1
preferred set
Do these preferences satisfy convexity
and nonsatiation?
Convexity yes
Nonsatiation yes
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x2
0 x1
preferred set
Do these preferences satisfy convexity
and nonsatiation?
Convexity yes
Nonsatiation yes
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x2
0 x1
preferred set
Do these preferences satisfy convexity
and nonsatiation?
Convexity yes
Nonsatiation yes
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x1x1
x2
0
B
A
C
6. Checking for convexity convex functions
With 2 goods and nonsatiation
the preferred set is convex
if the indifference curve
is a convex function.
A function is convex if any line
joining two points on its graph
lies entirely on or above the
graph.
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0 x
f(x)Facts about
convex
functions
A function is convex if any line joining two
points on its graph lies entirely on or below the
graph.
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0 x
f(x)Facts about
convex
functions
For functions with first and second derivatives
Convex functions are functions with increasing
first derivatives
Convex functions are functions with positive
second derivatives.
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x2
0 x1
A
Hicks assumed decreasing MRS
MRS = - gradient of indifference
curve
Hicks assumes that the MRS
decreases along an indifference
curve as x1 increases.
With 2 goods & nonsatiation
decreasing MRS is the same as
convexity.
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• Use the formula for the utility function to x2 as a function
of x1 & u.
• Find the second derivative of this function with respect to
x1.
• If the second derivative is positive the convexity
assumption is satisfied.
• With more than two goods checking for convexity
involves working with the second derivative matrix of
utility. Not for EC201.
How to check for convexity with two goods
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Example of checking for convexity
If u = x12/5x2
3/5 then
x2 = u5/3 x1-2/3 so
∂x2 = - (2/3) u5/3 x1-5/3 and
∂x1
∂2x2 = (10/9) u5/3 x1-8/3 > 0
∂x12
so the indifference curve is convex.
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Assumptions of Consumer Theory
1. Completeness
2. Transitivity
3. Continuity
4. Nonsatiation
5. Convexity
We have now
discussed all these.
We now look at some
implications.
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Now look at the implications of the
assumptions on:
• Crossing of indifference curves
• Finding the Marginal Rate of Substitution (MRS)
• Using the MRS to see if two utility functions represent
the same preferences
• Comparison of two people’s utility.
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Can indifference curves
cross?
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x2
x1
.A
.B
C
u(x1,x2) = 6
u(x1,x2) =4
A and B are on the same
indifference curve so are
B and C are on the same
indifference curve so are
Transitivity implies that A and C
are.
But A has utility than C.
7. Can indifference curves cross?
Transitivity
implies
indifference
curves
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x2
x1
.A
.B
C
u(x1,x2) = 6
u(x1,x2) =4
A and B are on the same
indifference curve so are
indifferent.
B and C are on the same
indifference curve so are
Transitivity implies that A and C
are
But A has utility than C.
Can indifference curves cross?
Transitivity
implies
indifference
curves
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x2
x1
.A
.B
C
u(x1,x2) = 6
u(x1,x2) =4
A and B are on the same
indifference curve so are
indifferent.
B and C are on the same
indifference curve so are
indifferent.
Transitivity implies that A and C
are
But A has utility than C.
Can indifference curves cross?
Transitivity
implies
indifference
curves
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x2
x1
.A
.B
C
u(x1,x2) = 6
u(x1,x2) =4
A and B are on the same
indifference curve so are
indifferent.
B and C are on the same
indifference curve so are
indifferent.
Transitivity implies that A and C
are indifferent.
But A has utility than C.
Can indifference curves cross?
Transitivity
implies
indifference
curves
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x2
x1
.A
.B
C
u(x1,x2) = 6
u(x1,x2) =4
A and B are on the same
indifference curve so are
indifferent.
B and C are on the same
indifference curve so are
indifferent.
Transitivity implies that A and C
are indifferent.
But A has higher utility than C.
Can indifference curves cross?
Transitivity
implies
indifference
curves
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x2
x1
.A
.B
C
u(x1,x2) = 6
u(x1,x2) =4
A and B are on the same
indifference curve so are
indifferent.
B and C are on the same
indifference curve so are
indifferent.
Transitivity implies that A and C
are indifferent.
But A has higher utility than C.
Can indifference curves cross?
Transitivity
implies
indifference
curves don’t
cross.
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Marginal Rate of
Substitution (MRS)
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8. Finding the marginal rate of substitution
• Definition and formula
• Explaining the formula
• Example: Cobb-Douglas utility function
• Using the MRS to find whether two utility functions
represent the same preferences
• Convexity and MRS
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x2
0 x1
A
Hicks and most textbooks defines
the MRS as
- gradient of indifference curve.
Perloff defines the MRS as
- gradient of indifference curve
Marginal Rate of Substitution: Definition
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∂u
∂x1
∂u
∂x2
0 x1
x2
Marginal Rate of Substitution: Formula
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x2
x2A u1
0 x1A x1B x1
u0
Explaining the formula for MRS
x2
x2B
x2A
If x2 stays at x2A
x1 changes from x1A to x1B
u changes from u0 to u1
Then
(u1 – u0) ≈ ∂u
(x1B - x1A) ∂x1
so u1 – u0 ≈ (x1B - x1A) ∂u
∂x1
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x2
x2A u1
0 x1A x1B x1
u0
Explaining the formula for MRS
x2
x2B
x2A
If x1 stays at x1A
x2 changes from x2A to x2B
u changes from u0 to u1
Then
(u1 – u0) ≈ ∂u
(x2B - x2A) ∂x2
so u1 – u0 ≈ (x2B - x2A) ∂u
∂x2
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x2
x2A u1
0 x1A x1B x1
u0
Explaining the formula for MRS
x2
x2B
x2A
From the previous slides
(u1 – u0) ≈ (x1B - x1A) ∂u
∂x1
≈ (x2B - x2A) ∂u
∂x2
So slope of line CD =
(x2B - x2A) ∂u
≈ ∂x1
(x1B - x1A) ∂u
∂x2
D
C
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x2
x2A u1
0 x1A x1B x1
u0
Explaining the formula for MRS
x2
x2B
x2A
When C and D are close the line
CD is close to a tangent
Gradient of the indifference
curve = - MRS
D
C
∂u
MRS = ∂x1
∂u
∂x2
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MRS example: Cobb-Douglas utility
function
utility u(x1,x2) = u = x12/5x2
3/5
MRS
1
2
52
2
52
1
53
2
53
1
2
1
3
2
5
35
2
x
x
xx
xx
x
u
x
u
//
//
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Using the MRS to find whether two utility
functions represent the same preferences
If f is a strictly increasing function and û(x1,x2) = f(u(x1,x2)) then
u and û represent the same preferences.
What if you have two utility functions and cannot see whether
they represent the same preferences?
Same preferences imply same indifference curves imply same
MRS.
Check whether utility functions they have the same MRS to
see whether they represent the same preferences..
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MRS from utility function u(x1,x2)
∂u
∂x1
∂u
∂x2
Using the chain rule MRS from utility function
û(x1,x2) = f(u(x1,x2)) ∂û df ∂u ∂u
∂x1 = du ∂x1 = ∂x1
∂û df ∂u ∂u
∂x2 du ∂x2 ∂x2
Using the MRS to find whether two utility
functions represent the same preferences
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Questions on preferences
and utility
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• Consumer theory does not give a way of doing this.
Can you compare different people’s utility?
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• Consumer theory does not give a way of doing this.
• Most policy changes are not Pareto improving.
Can you compare different people’s utility?
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• Consumer theory does not give a way of doing this.
• Most policy changes are not Pareto improving.
• Ethical and political value judgements matter.
Can you compare different people’s utility?
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Regret
Have you ever chosen to do or consume something and
then regretted it?
Choice under uncertainty?
Inability to commit yourself?
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How far does our welfare depend on
• relationships with other people?
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How far does our welfare depend on
• relationships with other people?
• What we think of ourselves?
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How far does our welfare depend on
• relationships with other people?
• What we think of ourselves?
• What other people think of us?
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How far does our welfare depend on
• relationships with other people?
• What we think of ourselves?
• What other people think of us?
• What we think other people think of us?
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Questions on preferences
• Are people rational?
• Who is the consumer?
• What is the effect of other consumers on your choices?
• What is the relationship between utility, welfare and happiness?
• What is the effect of what we buy and sell on our quality of life?
• Should we take psychology seriously?
• Is the simple model useful?
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Where we have got to?
• We have thought critically about the assumptions
(completeness, transitivity, continuity, nonsatiation &
convexity)
• we have got ordinal utility & indifference curves from the
assumptions
• We can use calculus to
• check for nonsatiation & convexity
• find the MRS
• see if two different utility functions represent the same
preferences.