Example of Duality1
From Problem Set 1, Q.2
Consider the following consumer problem:
max x1 x2 s.t. p1 x1 + p2 x2 6 M
(x1 ;x2 )
Uncompensated (‘Marshallian’) Demands, x (p; M ) - Holding income constant
Set the Lagrangian and derive the First Order Conditions (FOC). By using the equations
corresponding to x1 and x2 you should get the condition for the optimum M RS1;2 = p1 =p2 .2
In our exercise, it reads:
p1
u0
p1
x2
= 10 )
=
(1)
p2
u2
p2
x1
Then, you may plug the last result into the FOC that corresponds to
Marshallian (uncompensated) Demands:
x1 =
M
p1
+
and x2 =
M
p2
and obtain the
+
Trick 1: In interior solutions such as those obtained from Cobb-Douglas utility functions the condition MRS 1;2 =p 1 /p 2 always holds. Hence, you do not need
to derive the FOC (at least you are explicetly asked to do so). This equality and
the budget constraint are su¢ cient to …nd x .
Once we have x , it is possible to write down the indirect utility function just by plugging
it into the original utility function:
V (p1 ; p2 ; M ) =
M
+
+
p1
p2
(2)
Compensated (‘Hicksian’) Demands, h (p; U ) - Holding utility constant
Reverse the original problem and present it like a expenditure minimization problem. As a
constraint, set a minimum level of utility U :
min p1 x1 + p2 x2 s.t. x1 x2 > U
(x1 ;x2 )
One way of solving a minimization problem like this is to set up the usual Lagrangian and
change the sign of the constraint. Then, you can derive the three FOCs and check that the
condition in (1) holds. This should already give you taste of duality. Then, plug this result
1
2
It is very likely that there are typos. If you note them, please let me know and I will correct them.
u0
Recall that M RSi;;j = u0i .
j
1
into the FOC corresponding to
are the following:
h1 = U
and obtain h in the same way you did it with x . They
1
+
p2
p1
+
and h2 = U
p1
p2
1
+
+
Note Trick 1 can also be used to …nd h . The only di¤erence is that now you need to use
the new constraint: x1 x2 = U .
Finally, it is possible to work out the expenditure function in a process analogous to the one
that allowed us to …nd the indirect utility. Particularly, plug h into p1 x1 + p2 x2 and after a
bit of tedious algebra, you will obtain:
e(p1 ; p2 ; U ) = U
1
+
( + )
p2
p1
Trick 2: Thanks to duality we don’t need h to obtain e(p; U ). You can simply
invert the indirect utility function in (2): Substitute M by e(p; U ) and V (p; M )
by U and isolate e(p; U ) and you will see how much work you can save!
Slutsky Equation
The Slutsky Equation allows us, among other things, to derive the Income E¤ect (IE) and
Substitution E¤ect (SE) of a price change on the demand for either the same good or a
di¤erent good. The equation reads:
@hi
@xi
=
@pj
@pj
@xi
x
M j
The …rst term of the Left Hand Side (RHS) accounts for the SE while the second term
re‡ects the IE. Such result is quite relevant since it allows us to di¤erentiate between the
two e¤ects. Note that this would not be possible only by looking at the marginal change of
an uncompensated demand with respect to a price change as it only allows to observe the
sum of the two e¤ects, i.e.@xi =@pj = IE + SE. To get a better intuition, we can work this
out in our example.
LHS:
@x1
=
@p1
M
p21
+
RHS:
i) SE:
@h1
=
@p1
1
p1
+
2 +
+
U
1
+
p2
+
Because at the optimum we know that U = V (p1 ; p2 ; M ) holds. Hence, plug the
indirect utility function into the last result. Then you should obtain:
@h1
=
@p1
2
M
( + )2 p21
ii) IE:
@x1
1
x1 =
M
p1
M
p1
+
+
M
= 2
p1
2
+
Check yourself that by summing up SE and IE one obtains the result of the LHS...amazing!
3