?
?? + ?
?
??
(A4.10)
Substituting from (A4.5) into (A4.9), we get
Τ
?𝑈 ?𝐼 = ??
?
(
Τ
?? ?𝐼) + ??
?
(
Τ
?? ?𝐼) = ?
Τ
?
?
?? + ?
?
?? ?𝐼
(A4.11)
Substituting from (A4.10) into (A4.11), we get
Τ
?𝑈 ?𝐼 = ?
Τ
(?
?
?? + ?
?
?? ?
?
?? + ?
?
??) = ?
(A4.12)
Thus the
Lagrange multiplier
is the extra utility that results from an extra dollar
of income.

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54 of 50
An Example
●
Cobb-Douglas utility function
Utility function
U
(
X
,
Y
) =
X
a
Y
1−
a
,
where
X
and
Y
are two goods and
a
is a constant.
𝑈
?, ?
= ?log
?
+
1 − ? log(?)
The
Cobb-Douglas utility function
can be represented in two forms:
and
To find the demand functions for X and Y, given the usual budget constraint,
we first write the Lagrangian:
Φ = ?log
?
+
1 − ? log
?
− ?(?
?
? + ?
?
− 𝐼)
Now differentiating with respect to X, Y, and l and setting the derivatives equal
to zero, we obtain
Τ
𝜕Φ
𝜕?=
Τ
? ? − ??
?
= 0
Τ
𝜕Φ
𝜕Y=
Τ
(1 − ?) ? − ??
?
= 0
Τ
𝜕Φ
𝜕λ=?
?
X+?
?
? − 𝐼 = 0
𝑈
?, ?
= ?
𝑎
?
1−𝑎

55 of 50
Combining these expressions with the last condition (the budget constraint)
gives us
or
? =
Τ
1 𝐼.
Now we can substitute this expression for
λ
back into (A4.13) and
(A4.14) to obtain the demand functions:
The first two conditions imply that
?
?
? =
Τ
? ?
?
?
? =
Τ
(1 − ?) ?
(A4.13)
(A4.14)
Τ
? ? +
Τ
(1 − ?) ? − 𝐼 = 0
? = ( Τ
? ?
?
)𝐼
? = [(
Τ
1 − ?)
?
?
]𝐼

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56 of 50
Duality in Consumer Theory
●
duality
Alternative way of looking at the consumer’s utility
maximization decision: Rather than choosing the highest indifference curve,
given a budget constraint, the consumer chooses the lowest budget line that
touches a given indifference curve.
Minimizing the cost of achieving a particular level of utility:
Minimize
?
?
? + ?
?
?
subject to the constraint that
𝑈
?, ?
= 𝑈
∗
The corresponding Lagrangian is given by
Φ = ?
?
? + ?
?
? − ?(𝑈
?, ?
− 𝑈
∗
)
Differentiating
with respect to
X
,
Y
, and
μ
and setting the derivatives equal to
zero, we find the following necessary conditions for expenditure minimization:
(A4.15)
?
?
− ?MU
?
?, ?
= 0
and
?
?
− ?MU
?
?, ?
= 0
𝑈
?, ?
= 𝑈
∗