Mathematics for Economists
Professor Sang-Seung Yi
Fall 2016, Problem Set #4
1.
Find the eigenvectors and eigenvalues of D =
 2 2 

 . Based on the eigenvalues, what can you
 2  4
conclude about the sign of the quadratic form q = u’Du, where u’ = (u, v)?
D  I 
2
2
2
4
 (2   )(4   )  4  0
  3  5
Eigen value가 모두 음수이므로 q는 negative definite.
2. Given Q  L K  ( L  0, K  0) as a production function, find its Hessian matrix H. If   0 ,   0 ,
    1 , show that H is negative definite. In other words, Q is a strictly concave function.
QL  L 1K  , QK   L K  1
QLL   (  1) L  2 K  , QKK   (   1) L K   2 , QLK  QKL  L 1K  1 이므로
Q
H   LL
QLK
QLK  α( α  1 )Lα 2K β

QKK   αβLα 1 K β 1
 2
H1  QLL   (  1) L
αβLα 1 K β 1
β( β  1 )L K
α
β 2




K 0
H  QLLQKK  QLK QkL  αβ( 1  α  β )L2α 2K 2β 2  0
3. Find all the eigenvalues of the matrices and also the eigenvectors associated with the real eigenvalues
a)
0 6
 0
1 / 2 0 0


 0 1 / 3 0
−𝑟
1
|
|𝐷 − 𝑟𝐼| = | 2
0
r=1
−x + 6z = 0
1
( )x − y = 0
2
1
( )y − z = 0
3
x = 6z, y = 3z
0
−𝑟
1
3
6
0|
3
| = −𝑟 + 1 = 0
−𝑟
6
√46
𝑥
3
[𝑦] =
√46
𝑧
1
[√46]
 5  6  6
 1 4
b)
2 

 3  6  4
5−𝑟
|𝐷 − 𝑟𝐼| = | −1
3
r1 = 1 , r2 = 2
−6
4−𝑟
−6
−6
2 | = −(𝑟 − 2)2 (𝑟 − 1) = 0
−4 − 𝑟
𝑊ℎ𝑒𝑛 𝑟1 = 1
3
√19
𝑥
−1
[𝑦] =
√19
𝑧
3
[√19]
When r2 = 2,
3x − 6y − 6z = 0
−x + 2y + 2z = 0
3x − 6y − 6z = 0
when y = 0, x = 2z
when z = 0, x = 2y
𝑥
2𝑐1 + 2𝑐2
2
2
𝑐1
[𝑦] = [1] c1 + [0] 𝑐2 = [
]
𝑧
𝑐2
0
1
4. Find stationary/critical value of each function. Then, using second order condition, determine if it’s
maximum or minimum and calculate its maximum/minimum value.
a)
f ( x)  2 x 2  8 x  25
f ( x)  2 x 2  8 x  25
f ' ( x)  4 x  8 
x *  2 이고
f ' ' ( x)  4  0 이므로 극대값 f (2)  33 을 갖는다.
b)
f ( x)  x 3  3x 2  9 x  3
f ( x)  x 3  3x 2  9 x  3
f ' ( x)  3x 2  6 x  9 
f ' ' ( x)  6 x  6 
f ' ' (1)  12  0, f ' ' (3)  12  0 이다.
f (1)  8 , 극소값 f (3)  24 를 갖는다.
그러므로 극대값
5.
x*  1, 3 이고
An economy consists of two consumers with labels
i  1, 2 . They exchange two goods, labelled
j  1, 2 . Suppose there is a fixed total endowment e j of each good to be distributed between two
consumers. Let
c ij denotes i’s consumption of good j. Suppose that each consumer i has preferences
represented by the utility function
U i (c1i , c2i )   1 ln c1i   2 ln c2i
where the parameters  j are positive, and independent of i, with  1   2  1 . Suppose the goods are to
be distributed in order to maximize social welfare in the form of the weighted sum
W  1U 1   2U 2
, where the weights  i are positive, and 1   2  1 .
a) Formulate the welfare maximization problem with one equality constraint for each of the goods.
b) Write down the Lagrangian, where  j denotes the Lagrange multiplier associated with the constraint
for good j. Find the welfare maximizing distribution of the goods.
c) Verify that  j  W * e j , where W * denotes the maximum value of W.
Answer) TA 세션 자료 참조.
6. Prove that  is an eigenvalue of the matrix A if and only if  is an eigenvalue of A‘.
|𝐴 −  I| = 0
𝐴𝑠 |𝐴| = |𝐴′|,
𝐴𝑠 (𝐴 + 𝐵)′ = (𝐴′ + 𝐵′ ),
|𝐴′ −  I| = 0
′
|(𝐴 −  I) | = 0
′
|𝐴′ − (  I) | = 0