Repetition of
Consumer theory
Lectures in Microeconomic Theory
Fall 2007, Part 6
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G.B. Asheim, ECON4230-35, #6
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Consumer theory ― basic concepts
„
Preferences ― utility function
„
Indirect utility function
Property: Quasiconvex in good prices
„
„
Roy’s identity determines
the Marshallian demand function
Expenditure function
Property: Concave in good prices
Shephard’s lemma determines
the Hicksian demand function
The Slutsky equation
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G.B. Asheim, ECON4230-35, #6
Preferences
2
Completeness
x2
Transitivity
Continuity
{y | y
x}
Local nonsatiation
Monotonicity
x
Convexity
{y | y ≺ x}
x1
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G.B. Asheim, ECON4230-35, #6
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1
Utility function
„
From preferences to utility function
„
From utility function to better-off (and worse-off) sets
u ( y ) > u ( x) if and only if y
{y | y
„
„
x} = {y | u ( y ) > u ( x)}
The marginal rate of
dx2
substitution (MRS) MRS = − dx1
The elasticity of
substitution
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x
=
u = u ( x1 , x2 )
∂u
∂x1
∂u
∂x2
( )
x
MRS d x12
x2
dMRS
x1
σ=
u =u ( x1 , x2 )
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G.B. Asheim, ECON4230-35, #6
Utility functions ― Examples
„
Cobb-Douglas utility fn.
u ( x1 , x2 ) = x1a ⋅ x2b , 0 < a, b < 1, a + b = 1
„
Constant-elasticty-of-substitution (CES) utility fn.
(
σ −1
„
)
σ
σ −1 σ −1
u ( x1 , x2 ) = ax1 σ + bx2σ
, σ > 0, σ ≠ 1
Leontieff utility fn.
(ax1 , bx2 ),ofasubstitution?
u (What
x1 , x2 )is= the
minelasticity
> 0, b > 0
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G.B. Asheim, ECON4230-35, #6
Utility maximization
Assume price-taking behavior in good markets.
Indirect utility function : v (p, m) = max u (x)
x
such that px ≤ m
where m is income and p = ( p1 ,… , pn ) are good prices
Marshallia n demand fn : x(p, m) = arg max u (x)
x
such that px ≤ m
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G.B. Asheim, ECON4230-35, #6
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2
Indirect utility fn.
v (p, m) = max u ( x)
x
such that px ≤ m
Properties:
(1) Non - increasing in p, non - decreasing in m.
(2) Homogene ous of degree 0 in (p, m) :
v (tp, tm) = v (p, m) for all t ≥ 0.
x2
(3) Quasiconve x in p; that is
{p | v (p, m) ≤ k } is convex for all k
(4) Continuous in (p, m)
for positive prices and income
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x1
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G.B. Asheim, ECON4230-35, #6
Roy’s Identity
v (p, m) ≡ u ( x(p, m))
∂v (p, m)
∂pi
xi (p, m) = −
∂v (p, m)
∂m
provided the right-hand side is well-defined.
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G.B. Asheim, ECON4230-35, #6
Expenditure minimization
Assume price-taking behavior in good markets.
Expenditur e function : e(p,u ) = min px
x
such that u ( x) ≥ u
Hicksian demand function :
h (p,u ) = arg min px
x
such that u ( x) ≥ u
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G.B. Asheim, ECON4230-35, #6
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3
What happens when a good price increases,
while utility is constant? x2
p′x′ ≤ p′x′′
p′′x′′ ≤ p′′x′
p′(x′ − x′′) ≤ 0
( x1′, x′2 )
− p′′(x′ − x′′) ≤ 0
( x1′′, x2′′ )
( x1′′, x′2′)
If p′j = p′j′ for all j ≠ i , then
( pi′ − pi′′)(xi′ − xi′′) ≤ 0
x1
pi′′ > pi′ implies xi′′ ≤ xi′
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G.B. Asheim, ECON4230-35, #6
Expenditure fn.
e(p, u ) = min px s.t. u ( x) = u
x
Properties:
(1) Non - decreasing in p.
(2) Homogene ous of degree 1 in p :
e(tp, u ) = te(p, u ) for all t ≥ 0.
e( pi , p′−i , u )
(3) Concave in p
Slope : hi ( pi′, p′−i , u )
e( pi , p′−i , u )
e( pi′, p′−i , u )
(4) Continuous in p
p′i
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pi
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G.B. Asheim, ECON4230-35, #6
Shephard’s Lemma
e( pi , p′−i , u )
Slope : hi ( pi′, p′−i , u )
e( pi , p′−i , u )
e( pi′, p′−i , u )
e(p, u ) ≡ ph (p, u )
p′i
Assume differentiability.
hi (p, u ) =
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∂e(p, u )
∂pi
pi
Invoke concavity.
∂hi (p, u ) ∂ 2 e(p, u )
=
≤0
∂pi
∂pi2
G.B. Asheim, ECON4230-35, #6
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4
Important identities
max u ( x)
min px
x
x
s.t. u ( x) ≥ u
s.t. px ≤ m
↓
v (p, m)
↓
e(p,u )
u = v(p, e(p, u ))
m = e(p, v(p, m))
Shephard’s
lemma
Roy’s
indentity
x(p, m) =
h(p, v (p, m))
h (p , u ) =
x(p, e(p, u ))
x(p, m)
h (p , u )
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G.B. Asheim, ECON4230-35, #6
The Slutsky equation
h (p, u ) = x (p, e(p, u ))
∂h j (p, u )
∂pi
=
∂x j (p, e(p, u ))
∂pi
+
∂x j (p, e(p, u )) ∂e(p, u )
∂m
∂pi
By invoking Shephard’s lemma:
∂x j (p, m)
∂pi
Total effect
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=
∂h j (p, u )
∂pi
−
∂x j (p, m)
∂m
Substitution effect
xi (p, m)
Income effect
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G.B. Asheim, ECON4230-35, #6
Revealed preferences x
p′x′ ≥ p′x′′
x′ is chosen when x′′ is affordable
u ( x′) ≥ u (x′′)
x′ is directly revealed preferred to x′′
2
( x1′′, x2′′ )
Weak axiom of
revealed preferences
( x1′, x′2 )
x1
If x′ is directly r evealed preferred to x′′
and x′ ≠ x′′, then it is not the case that
x′′ is directly r evealed preferred to x′.
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