Some Duality Results
INDIRECT UTILITY FUNCTION
The indirect utility function is defined as the maximum utility that
can be attained given money income and goods prices.
u^{*}(p_{1},p_{2},M)
= max U(x_{1},x_{2}) s.t. p_{1} x_{1} + p_{2} x_{2} = M
Properties of the indirect utility function:
 u^{*} is decreasing in prices and
increasing in income
 u^{*} is
homogeneous of degree 0 in prices and income
 u^{*} is quasiconvex
in prices
 Roy's identity: x_{i}^{*} =
(∂u^{*}/∂p_{i})
/
(∂u^{*}/∂M)
The first two properties are obvious.
To see why u^{*} is quasiconvex, consider
the following diagram.
Let x_{1}^{0} and
x_{2}^{0} be the
optimal demand bundle associate with prices
(p_{1}^{0},
p_{2}^{0}) and income M. Let
U(x_{1}^{0},
x_{2}^{0})=u. Therefore
u^{*} at the point
(p_{1}^{0},
p_{2}^{0}) equals u.
Consider some other point on the
straight line. When goods prices fall on this straight line, the original
goods
bundle is still affordable. Thus the consumer can attain at least a utility
of at least u. But typically, the optimal choice of goods will allow a
utility level higher than u. If we draw an "indifference curve" on the price
space, therefore, such an indifference curve will lie above the straight line.
Prices to the northeast of the indifference curve are worse than prices on
the indifference curve. From the diagram, we conclude that the lower
contour sets are convex. This is precisely the definition of quasiconvexity.
The fourth property, Roy's identity, is a consequence of the envelope
theorem. Since the Lagrangian of the problem is
U(x_{1},x_{2})+
λ(M  p_{1} x_{1} 
p_{2} x_{2})
We have
∂u^{*}/∂p_{1}
=
 λ^{*}
x_{1}^{*} and
∂u^{*}/∂M =
λ^{*}.
Dividing one equation by the
other yields Roy's identity.
EXPENDITURE FUNCTION
The expenditure function is defined as the minimum expenditure required to
attain a utility level u, given goods prices. That is,
C^{*}(p_{1},
p_{2}, u) =
min p_{1} x_{1} +
p_{2} x_{2} s.t.
U(x_{1}, x_{2})=u
You can see that the expenditure function is formally equivalent to the cost
function introduced in producer theory.
All their mathematical properties are
the same, so you can refer back to the earlier notes.
It is also clear that you can derive the cost function from the indirect
utility function, and vice versa. If the minimum cost of achieving utility
level u is M, then the maximum utility from income M is u. The cost function
and indirect utility function are therefore inverses of each other. We have
C^{*}( p_{1},
p_{2},
u^{*}(p_{1},
p_{2}, M) )
≡ M
and
u^{*}( p_{1},
p_{2},
C^{*}(p_{1},
p_{2}, u) )
≡ u
DIRECT AND INDIRECT UTILITY FUNCTIONS
Given an indirect utility function
u^{*}(p_{1},
p_{2}, M), we can find the (direct)
utility function U(x_{1},
x_{2}) by this relationship:
U(x_{1}, x_{2}) =
min_{p1,
p2}
u^{*}(p_{1},
p_{2}, M) s.t.
p_{1} x_{1} +
p_{2} x_{2} = M
To see why this is true. Let
x=(x_{1}, x_{2})
be the demanded bundle when prices are
p=(p_{1}, p_{2}) and income is
M. Then, by definition, u^{*}(p, M) = U(x). Let p' be
any other price vector that satisfies p'x=M. Since x is still affordable, we
must have
u^{*}(p',M) ≥ U(x) for all p' x = M
So U(x) is indeed the solution to the minimization problem.
SLUTSKY EQUATION
Start off with a Marshallian demand
x_{1}=
x_{1}^{*}(
p_{1}, p_{2}, M).
Let utility at this
demand bundle be u. When p_{1} changes,
holding M constant, the level of utility will change.
Suppose, now, when p_{1} changes,
M is also changed to keep utility constant at
u, then we get a Hicksian demand curve,
x_{1}=
x_{1}^{~}
(p_{1}, p_{2}, u).
To keep utility constant at u, money income has to be equal to
C^{*}(p_{1},
p_{2}, u).
Therefore we may write
x_{1}^{*}(
p_{1},
p_{2},
C^{*}(p_{1},
p_{2},u) ) ≡
x_{1}^{~} (
p_{1},
p_{2}, u )
Differentiate both sides with respect to p_{1},
∂x_{1}^{*}/
∂p_{1}
+
(∂x_{1}^{*}/∂M)
(∂C^{*}/∂p_{1}) =
∂x_{1}^{~}
/∂p_{1}
Note that
∂C^{*}/∂p_{1} =
x_{1}^{~} =
x_{1}^{*}. Therefore
∂x_{1}^{*}/
∂p_{1} +
x_{1}^{*}(
∂x_{1}^{*}/∂M) =
∂x_{1}^{~}/
∂p_{1}
Since we already know that the right hand side is negative, we conclude that
the left hand side is also negative. Alternatively, we can write
∂x_{1}^{*}/
∂p_{1} =
∂x_{1}^{~}/
∂p_{1} 
x_{1}^{*}
(∂x_{1}^{*}/∂M)
The first term on the right hand side is always negative, and we call it the
substitution effect. The second term is the income effect, and
it may be either positive or negative.

Lecture Notes

Consumer Theory
Consumer
surplus