Assignment 3
Problem 4.
a) Varian, chapter 7.
b) Use the Slutsky equation:
@xj
@xj
@hj
=
+
xi ; i; j 2 f1; 2g
@pi
@pi
@m
and show that
@x1
@p2
= 0:2 and
@x2
@p1
=
0:1 (you have to use the fact that
@h1
@p2
=
@h2
@p1 ).
c) Recall the Slutsky equation when income is not …xed, but determined by the endowments ! 1 and ! 2 .
@hj
@xj
@xj
=
+
(xi ! i ) ; i; j 2 f1; 2g
@pi
@pi
@m
Use it to show that
@x1
@p2
= 0:2 and
d) The correct de…nition is that
de…nition is that we may have
@xi
@pj
@x2
@p1
@hi
@pj
= 1:1.
< 0, since
<0<
@hj
@pi
=
@hi
@pj .
The problem with the alternate
@xj
@pi .
Problem 5.
a) Varian, chapter 7.
b) It is a standard Cobb-Douglas utility function. To …nd the Marshallian demand
functions maximize utility over the budget set and you should get
am
x0 (p0; p; m) =
(0.1)
,
p0
bm
x (p0; p; m) =
.
p
It is easy to see that the expenditure on good 1, (p0 x0 = am), and on good 2, (px = bm),
are independent of p0 and p.
To …nd the Hicksian demand, minimize the expenditure p0 x0 +px subject to u (x0 ; x) u.
(Also, one can use the Shephard’s lemma)
The Hicksian demand functions are
(0.2)
ap
bp0
b
h0 (p0; p; u) =
bp0
ap
a
h (p0 ; p; u) =
u,
u.
c) The Marshallian demand functions are the solution of the following utility maximization problem
max xa0 (min fx1 ; x2 g)b subject to p0 x0 + p1 x1 + p2 x2
1
m.
2
First, argue that at the optimal we must have x1 = x2 (recall the argument that we used
while solving Leontief utility function).
Given that x1 = x2 (call it x), the above utility maximization problem can be rewritten
as
max xa0 xb subject to p0 x0 + (p1 + p2 ) x m.
Denoting p1 + p2 by p, this problem reduces to the utility maximization problem in question
(b).
With similar arguments show that the expenditure minimization problem in (c) reduces
to the problem in (b) by setting x1 = x2 = x and p1 + p2 = p.
d) By setting x1 = x2 = x and p1 + p2 = p, the Hicksian demand functions for the three
goods case can be written directly from equations (0.1).
am
x0 (p0; p1 ; p2 ; m) =
,
p0
bm
,
x1 (p0; p1 ; p2 ; m) =
p1 + p2
bm
x2 (p0; p1 ; p2 ; m) =
.
p1 + p2
Use Slutsky Lemma to derive the terms of the substitution matrix. You can check that the
diagonal terms and cross price e¤ects between goods 1 and 2 are negative.