Econ 600
Schroeter
Fall 2008
Exam #1 Solution Outline
1. a. False. Change negative definite to negative semi-definite.
b. False. There are several ways to correct it with small changes. One way:
If F () is strictly concave then it is both strictly quasi-c
Econ 600
Schroeter
Fall 2009
Homework Solution
1. For the long-run problem:
(
)
(
)
*
*
p f1 x1 , x 2 w1 = 0
*
*
p f 2 x1 , x 2 w2 = 0
Note that the assumption of a negative definite Hessian for f () insures that the Jacobian
of the system of equations is
Econ 600
Schroeter
Fall 2008
Homework Solution Outline
1.
min
w.r .t . y , w1 , w2 , x1 , x2
F ( y; w1 , w2 ; x1 , x 2 ) subject to f ( x1 , x 2 ) y = 0
where F ( y; w1 , w2 ; x1 , x 2 ) C ( x1 , x 2 ; w1 , w2 ) C * ( y; w1 , w2 ) and
C ( x1 , x 2 ; w1 ,
Econ 600
Schroeter
Fall 2011
Exam #2 Solution Outline
1. a. I intended the answer to this one to be True. But the conclusion actually requires
that * 0 . So I gave credit either for True or for False with the correction noted.
b. False. Change quasi-conca
Econ 600
Schroeter
Fall 2010
Exam #2 Solution Outline
1. a. False. Change max to min.
d. False. Change to .
b. True.
c. True.
e. True.
2. Proof:
Given > 0 , want to show that there exists N such that n N y n a < .
By convergence of cfw_x n n =1 and cfw_z
Econ 600
Schroeter
Fall 2009
Exam #2 Solution Outline
1. a. False. Change "if and only if" to "only if" or change concave to quasiconcave
b. True.
c. True.
d. True.
e. False. Change "negative definite" to "negative semi-definite".
2. a.
max U ( x1 + x 2 ,
Econ 600
Schroeter
Fall 2008
Exam #2 Solution Outline
1. a. True.
b. True.
c. False. Change
d2 *
2
F (a 0 ) 2 F x * (a 0 ); a 0
da 2
a
d. False. Change
F *
F *
x = 0 to
x 0.
xi
xi
(
()
)
to
d2 *
2
F (a 0 ) 2 F x * (a 0 ); a 0 .
da 2
a
(
)
()
e. False. One
Econ 600
Schroeter
Fall 2011
Exam #1 Solution Outline
1. a. True.
b. False. Change if and only if to if.
c. False. Change positive semi-definite to positive definite. (And then could also
strengthen the claim by changing local minimum to strict local mini
Econ 600
Schroeter
Fall 2010
Exam #1 Solution Outline
1. a. True.
b. True.
d. False. Change "
c. True.
2
d2
2
d2
F x * (a ); a 2 F * (a ) " to " 2 F x * (a ); a 2 F * (a ) ."
a 2
da
a
da
(
)
(
)
e. False. Change
"
d*
d*
F (a ) =
F x * (a ); a " to "
F (a
Econ 600
Schroeter
Fall 2009
Exam #1 Solution Outline
1. a. True.
b. False. Change "if and only if" to "if".
c. False. Change "strictly quasi-concave" to "strictly concave".
d. True.
e. Strictly speaking, this is false as written. In order for the stateme
Econ 600
Schroeter
Fall 2010
Homework solution outline
1. a.)
py w1 x1 w2 x 2 subject to f ( x1 , x 2 ) y = 0
max
w. r .t . y , x1 , x2
Lagrangian: L( y, x1 , x 2 ; ) = py w1 x1 w2 x 2 + ( y f ( x1 , x 2 )
FONC:
L
*
(*) = y * f x1* , x2 = 0
(
)
L
(*) = p