Econ 0702 Mathematical Methods in Economics
Answer Keys for Problem Set 2
1.1
FOC:
dy
 10 x  1
dx
10 x  1  0
x* 
SOC:
1
1
, y* 
10
20
d2y
 10  0
dx 2
 ( x* 
1
1
, y*  ) is a relative minimum.
10
20
1.2
FOC:
dy
 3x 2  3
dx
3x 2  3  0
x*  1or  1(rejected )
y*  3
SOC:
d2y
 6x
dx 2
If x  1,
d2y
60
dx 2
 ( x*  1, y*  3) is a relative minimum.
1.3
FOC:
dy
2x

 0 for any values of x ; there exists no relative extremum.
dx (1  2 x) 2
1
2a)
 ( x)  63  98x  62 x 2  18x 3  2 x 4  R4
2b.1)
FOC:
dy
 4x 3
dx
 4x3  0
x*  0, y*  0
SOC:
f ' ' ( x )  12 x 2
f ' ' ( 0)  0
f ' ' ' ( x )  24 x
f ' ' ' ( 0)  0
f 4 ( x )  24  0
 The first non-zero derivative value is f 4  24 .
 ( x*  0, y*  0) is a relative maximum.
2b.2)
FOC:
dy
 3( x  1) 2
dx
3( x  1) 2  0
x*  1
2
SOC:
f ' ' ( x)  6( x  1)
f ' ' (1)  0
f ' ' ' ( x)  6  0
 The first non-zero derivative is f ' ' '.
 ( x*  1, y*  16) is an inflection point.
3ai)
 3  2  u 
 
q  u v 
  2 7  v 
3aii)
  1 2  3  u 

 
q  u v w 2  4 0  v 
  3 0  7  w 

 
3bi)
D1  3  0
D2  17  0
 q is positive definite.
3bii)
D1  1  0
D2  0
 q is neither positive definite nor negative definite.
3
4a)
The first-order condition:
the only solution is:
2
3
0
6
4
4
12
H  3
0
H1  2  0
H2  3  0
H3  4  0
 d 2 z is positive definite.
z*  0 is a minimum.
4b)
The first-order condition:
4
4 0
0
H 0 1
0
0 0 4e
H1  4  0
H2  4  0
H 3  16e  0
 d 2 z is positive definite.
 z*  2  e is a minimum.
5. (a)
5
6
5 (b) (i) The set of points lying on or above an exponential curve; a convex set.
160
140
120
100
80
60
40
20
0
1
2
3
4
5
6
7
8
9
10
(ii) The set of points lying on or above a rectangular hyperbola in the positive quadrant; a
convex set.
2.5
2
1.5
1
0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
7