ECON6021 Microeconomic Analysis

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ECON6021 Microeconomic Analysis

• Consumption Theory I

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Topics covered

• Budget Constraint
• Axioms of Choice Indifference Curve
• Utility Function
• Consumer Optimum

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Bundle of goods

• A is a bundle of goods consisting of XA units of
  good X (say food) and YA units of good Y (say
  clothing).
• A is also represented by (XA,YA)

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Convex Combination
y
(xA, YA)
A
C
(xA, YB)
B
x
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Convex Combination
? C is on the st. line linking A B
Conversely, any point on AB can be written as
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Slope of budget line
(market rate of substitution)
Unit
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Example jar of beer Px4 loaf of bread
Py2
Both Px and Py double,
No change in market rate of substitution
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Tax a 2 levy per unit is imposed for each good
? Slope of budget line changes
After doubling the prices
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Axioms of Choice

• Indifference Curve

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Axioms of Choice

• Nomenclature
• ? is preferred to
• ? is strictly preferred to
• ? is indifferent to
• Completeness (Comparison)
• Any two bundles can be compared and one of the
  following holds A?B, B ?A, or both (? AB)
• Transitivity (Consistency)
• If A, B, C are 3 alternatives and A?B, B ?C, then
  A ?C
• Also If A?B, B?C, then A ?C.

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Axioms of choice

• Continuity
• A?B and B is sufficiently close to C, then A ?C.
• Strong Monotonicity (more is better)
• A(XA , YA), B(XB , YB) and XAXB, YAYB with at
  least one is strict, then AgtB.
• Convexity
• If A?B, then any convex combination of A B is
  preferred to A and to B, that is, for all 0 ?t
  lt1,
• (t XA(1-t)XB, tYA(1-t)YB) ? (Xi , Yi), iA or
  B.
• If the inequality is always strict, we have
  strict convexity.

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Indifference Curve

• When goods are divisible and there are only two
  types of goods, an individuals preferences can
  be conveniently represented using indifference
  curve map.
• An indifference curve for the individual passing
  through bundle A connects all bundles so that the
  individual is indifferent between A and these
  bundles.

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Properties of Indifference Curves

• Negative slopes
• ICs farther away from origin means higher
  satisfaction

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Properties of Indifference Curves

• Non-intersection
• Two indifference curves cannot intersect
• Coverage
• For any bundle, there is an indifference curve
  passing through it.

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Properties of Indifference Curves

• Bending towards Origin
• It arises from convexity axiom
• The right-hand- side IC is not allowed

Y
X
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Utility Function
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Utility Function

• Level of satisfaction depends on the amount
  consumed UU(x,y)
• U0 U(x,y)
• All the combination of x y that yield U0 (all
  the alternatives along an indifference curve)
• yV(x,U0), an indifference curve
• ?U(x,y)/?x, marginal utility respect to x,
  written as MUx.

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(by construction)
(if strong monotonicity holds)
Slope
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Y
A
B
X
The MRS is the max amount of good y a consumer
would willingly forgo for one more unit of x,
holding utility constant (relative value of x
expressed in unit of y)
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• Marginal rate of substitution

DMRS
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Measurability of Utility
An order-preserving re-labeling of ICs does not
alter the preference ordering.
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Positive monotonic (order-preserving)
transformation

• They are called positive monotonic transformation

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Positive Monotonic Transformation

• What is the MRS of U at (x,y)?
• How about U?

?
?
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Positive Monotonic Transformation

• ICs of order-preserving transformation U
  overlap those of U.
• However, we have to make sure that the numbering
  of the IC must be in same order before after
  the transformation.

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Positive Monotonic Transformation

• Theorem Let UU(X,Y) be any utility function.
  Let VF(U(X,Y)) be an order-preserving
  transformation, i.e., F(.) is a strictly
  increasing function, or dF/dUgt0 for all U. Then V
  and U represent the same preferences.

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Proof

• Consider any two bundles and
• Then we have

Q.E.D.
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Consumer Optimum
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Constrained Consumer Choice Problem

• Preferences represented by indifference curve
  map, or utility function U(.)
• Constraint budget constraint-fixed amount of
  money to be used for purchase
• Assume there are two types of goods x and y, and
  they are divisible

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Consumption problem

• Budget constraint
• I0 given money income in
• Px given price of good x
• Py given price of good y
• Budget constraint I0?PxxPyy
• Or, I0 PxxPyy (strong monotonicity)
• dI0 PxdxPydy0 (by construction)
• Pxdx-Pydy

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Psychic willingness to substitute
At B, my MRS is very high for X. Im willing to
substitute XA-XB for YB-YD. But the market
provides me more X to point D! ?
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Consumer Optimum

• Normally, two conditions for consumer optimum
• MRSxy Px/Py (1)
• No budget left unused (2)

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Y
Both A C satisfy (1) and (2) Problem bending
toward origin does not hold.
U1
U0
A
C
X
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Special Cases
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Quantity Control

• Max UU(x,y)
• Subject to (i) I PxxPyy
• (ii) Rx

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1. Corner at x0
2. Interior solution 0ltxltR
3. corner at R
4. corner at R

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An Example U(x,y)xy
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A satisfies (1) but not (2) B, C satisfy (2) but
not (1) Only D satisfies both (1) (2)
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Other Examples of Utility Functions
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An application Intertemporal Choice

• Our framework is flexible enough to deal with
  questions such as savings decisions and
  intertemporal choice.

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Intertemporal choice problem
Income in period 2
u(c1,c2)const
C2
1600
500
Slope -1.1
C1
1000
Income in period 2
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• 1000-C1S (1)
• 500S(1r)C2 (2)
• Substituting (1) into (2), we have
• 500(1000-C1)(1r)C2
• Rearranging, we have
• 15001000r-(1r) C1C2 gt C
• Using C1C2C, we finally have

r ? ? C ? (S ?)