Economics 21: Intermediate Microeconomics
Topic 2: Consumer Theory – Duality Theory (i)
1
Economics 21: Intermediate Microeconomics
Topic 2:
Consumer Theory
Duality Theory (i)
Reference: Varian, p.93
Outline:
I.
Introduction
II.
CobbDouglass Preferences
I.
Introduction
We
explore ‘duality’ in the context of consumer theory. As we have seen, there are three steps to
consumer decisionmaking:
(i)
Preferences: i.e. what the individual
wants
to do.
(ii)
The Budget Constraint: i.e. what the individual
is able
do.
(iii)
Constrained Optimisation: i.e. how the individual reconciles (i) and (ii) by
endeavouring to reach the highest
feasible
level of satisfaction.
The consumer’s problem can be interpreted as either:
(a)
Maximising utility subject to an income constraint, yielding the
Marshallian
demand function; or
(b)
Minimising expenditure subject to a utility constraint, yielding the
Hicksian
(or
Compensated
) demand function.
Duality theory explores the ‘duality’ that exists between these two optimisation problems. It is
best explained by exploring the two optimisation problems in detail. We will firstly analyse a
general representation of utility and then [Duality Theory (ii)] explore a specific utility function
(the CobbDouglas function).
II.
Marshallian Demand Functions
The consumer’s problem is:
(
)
2
1
2
1
,
,
max
x
x
u
x
x
(1)
subject to:
m
x
p
x
p
=
+
2
2
1
1
(2)
The solutions is derived by maximizing the following Lagrange function:
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Economics 21: Intermediate Microeconomics
Topic 2: Consumer Theory – Duality Theory (i)
2
max
x
1
,
x
2
,
!
L
g
=
u x
1
,
x
2
(
)
+
!
m
"
p
1
x
1
"
p
2
x
2
(
)
(3)
The FOCs for maximization are given by:
0
1
1
1
=
!
=
p
u
x
L
x
g
"
#
#
(4)
!
L
g
!
x
2
=
u
x
2
"
#
p
2
=
0
(5)
0
=
!
=
px
m
L
g
"#
"
(6)
Dividing (4) by (5) yields the familiar result:
2
1
2
1
p
p
u
u
x
x
=
(7)
Equation (7) shows that in equilibrium, the consumer’s MRS between the two goods equals the
ERS between the two goods. Equation (6) states that the individual must exhaust his entire
budget. Thus equations (6) and (7)
together
imply the tangency between the consumer’s
indifference curves and his budget constraint.
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 Summer '09
 JOHNG.SESSIONS
 Microeconomics, p1

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