Math Camp
Economics/Resource Economics Departments
August 2008
Kambiz Raffiee
1.
Consider the following system of equations:
y2  z  u  v  w 3  1
 2x  y  z 2  u  v 3  w   3
x2  z  u  v  w3  3
The point P = (x, y, z, u, v, w) = (1, 1, 0, -1, 0, 1) is a solution. Apply the implicit
function theorem to prove that the system defines u, v, and w a continuously
differentiable functions of x, y, and z in a neighborhood of P. Find u 'x , v 'x , and w 'x at P.
2.
Consider the following separable utility function:
U( x 1 , x 2 )  U1 ( x 1 )  U( x 2 )
where
U i'
U i
 U i''  0. Show that neither good can be inferior, i.e.,
 U i'  0 and
x i
x i
x i
 0, where M is money income. (Hint: Set up the Lagrange function for constrained
M
utility maximization and apply the implicit function theorem to the first-order
conditions).
3.
Consider the following implicit market demand and market supply functions:
QD = D(P, M)
QS = S(P, C, t)
where QD = quantity demanded, P = product price, M = income, QS = quantity supplied,
C = cost index of inputs, and t = excise tax. The signs of the partial derivatives are DP <
0, DM > 0, SP > 0, SC < 0, and St < 0. Applying the market equilibrium condition of QD =
QS = Q, we get:
D(P, M) – Q = 0
S(P, C, t) – Q = 0
a.
Use the implicit function theorem to calculate the Jacobian determinant. Is it nonzero?
b.
Find
P
Q
and
. Determine their sign and provide an economic
M
M
interpretation.
c.
Find
P
Q
and
. Determine their sign and provide an economic
C
C
interpretation.
d.
Find
P
Q
and
. Determine their sign and provide an economic
t
t
interpretation?
4.
For each of the following functions, find the stationary point and determine
whether that point is a relative maximum, minimum, or saddle point.
c.
f (x1 , x 2 )  x12  4x1x 2  2x 22
f (x1 , x 2 )   4x1  6x 2  x12  x1x 2  2x 22
f (x1 , x 2 )  12x1  4x 2  2x12  2x1 x 2  x 22
5.
Consider the generalized Cobb-Douglas function:
a.
b.
z  Ax 1a1 x a22 ............................x ann .
a.
Compute the kth principal minors of the Hessian H(x) and prove that its value is:
a 1 1
a2
a 1 ......a k
Dk 
zk
2
( x 1 ......x k )
b.
.
.
.
ak
a1
...... ...... a 1
a 2  1 ............ a 2
. 
.
.
 .
.

a k ........
a k 1
Prove that:
 k
 a ......a k
D k  (1) k  1   a i 1 1
zk
2
 i 1
 ( x 1 ......x k )
c.
Prove that the function is strictly concave for
a
i
1.
6.
Suppose the optimal capacity utilization by a firm requires that its output
quantities x1 and x2, and capacity level k should be chosen to solve the problem:
 x1  k ,
 x k,
 2
max x 1  3x 2  x 12  x 22  k 2 subject to x 1  0 ,
x  0,
 2
k  0.
Show that k = 0 cannot be optimal, and then find the solution.