Problem set 1 1. Find extreme points of the following functions: (a

Problem set 1
1. Find extreme points of the following functions:
(a) f (x; y) = 3x2 + 3xy
(b) f (x; y) = 4xy +
1
x
y2
+
15x
1
y
(c) f (x; y; z) = x2 + y 2 + z 2
12xy + 2z
(d ) f (x; y) = xy ln(x2 + y 2 )
(e ) f (x; y) = x2 y 3 (6
x
y)
2. Classify the stationary points of
f (x; y; z) =
3
2x + 15x
2
36x + 2y
3z +
Z
z
2
et dt:
y
3. (a) Solve the utility maximizing problem (assuming m
4)
max
U (x; y) = 21 ln(x + 1) + 14 ln(y + 1)
subject to 2x + 3y = m
(b) With U (m) as the indirect utility function, show that
4. Find the extreme points of f (x; y) = x2
dU
dm
= :
y 2 on the unit circle x2 + y 2 = 1
(a) using the Lagrange multipliers method
(b) using the substitution y 2 = 1
unconstrained problem
x2 ; solve the same problem as a single variable
(c) Do you get the same results? Why or why not?
5. Answer the following questions about an individual with CES utility function
1
U (x1 ; x2 ) = (x1 + x2 ) with 0 <
<1
who maximizes utility subject to the standard budget constraint:
p1 x1 + p2 x2 = M:
(a) Set up the Lagrangian and solve the …rst-order conditions for the regular demand
functions x1 (p1 ; p2 ; M ) and x2 (p1 ; p2 ; M ):
(b) Are these demand functions homogeneous of any degree? (Show why or why not.)
(c) Find an expression for the optimal value of the Lagrange multiplier
(p1 ; p2 ; M ):
(d) Is the indirect utility function V (p1 ; p2 ; M ) = U (x1 (p1 ; p2 ; M ); x2 (p1 ; p2 ; M )) homogeneous of any degree? If so, what degree?
6. Find the extreme points of f (x; y; z) = 2x + 4y
1
4z subject to x2 + 16y 2 + 4z 2 = 16:
7. For the following functions …nd extreme points subject to the given constraints:
(a) f (x; y) =
s.t. x + y 2 = 1
xy
s.t. x
(b) f (x; y; z) = (x + y)z
(c) f (x; y) =
1
x
+
1
y
s.t.
1
x2
+
1
y2
2
+y
2
+ 2z
2
=4
1=0
8. Point x = (2; 1; 0) is an extreme point of f (x; y; z) = x + 2y z subject to
x2 + y 2 + z 2 = 5 and x y + z = 1: Is it true or not? Explain your answer.
2

Download Report