Econ 600
Schroeter
Fall 2004
Exam #1
Do all three problems. Weights: #1 - 25%, #2 - 35%, #3 - 40%. Closed book, closed
notes. Be sure your answers are presented in a neat and well-organized manner.
1. Answer True or False for each of the following stateme
Econ 600
Schroeter
Fall 2009
Exam #2
Do all four problems. They will be equally weighted. Closed book, open notes. Be sure
your answers are presented in a neat and well-organized manner.
1. Answer True or False for each of the following statements. If the
Econ 600
Schroeter
Fall 2010
Exam #2
Do all four problems. They will be equally weighted. Closed book, open notes. Be sure
your answers are presented in a neat and well-organized manner.
1. Answer True or False for each of the following statements. If the
Econ 600
Schroeter
Fall 2011
Exam #2
Do all four problems. Weights: #1 30%, #2 20%, #3 20%, #4 30%. Closed book,
open notes. Be sure your answers are presented in a neat and well-organized manner.
1. Answer True or False for each of the following statemen
Econ 600
Schroeter
Fall 2012
Exam #2
Do all four problems. They will be equally weighted. Closed book, open notes. Be sure
that your answers are thoroughly explained and presented in a neat and wellorganized manner!
1. Answer True or False for each of the
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 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 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 2012
Exam #2 Solution Outline
1. a. False. Change if and only if to only if.
c. False. Change
b. True.
*
dF *
(a1 ) = F x * (a1 ); a1 to dF (a1 ) = L x * (a1 ); * (a1 ); a1
da
a
da
a
(
)
(
)
where L() is the Lagrangian defined by
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 2013
Exam #1 Solution Outline
1. a. False. Change to If F : n is differentiable and
definite for all x * n , then F is strictly concave.
b. True.
2F *
x is negative
x 2
c. True
d. False. Change convex to strictly convex; OR change
Econ 600
Schroeter
Fall 2012
Exam #1 Solution Outline
1. a. True.
b. False. Change if and only if to only if.
c. True.
d. False. There are several ways to change it to a correct statement. One example:
If F : n is strictly concave, then it is also strictl
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
Exam #1
Do all three problems. Weights: #1 - 30%. #2 - 30%, #3 - 40%.
Closed book, closed notes. Be sure your answers are presented in a neat and wellorganized manner.
1. Answer True or False for each of the following statemen
Econ 600
Schroeter
Fall 2010
Exam #1
Do all three problems. Weights: #1 - 30%. #2 - 30%, #3 - 40%.
Closed book, closed notes. Be sure your answers are presented in a neat and wellorganized manner.
1. Answer True or False for each of the following statemen
Econ 600
Schroeter
Fall 2011
Exam #1
Do all three problems. Weights: #1 - 30%, #2 - 35%, #3 - 35%.
Closed book, closed notes. Be sure your answers are presented in a neat and wellorganized manner.
1. Answer True or False for each of the following statemen
Econ 600
Schroeter
Fall 2012
Exam #1
Do all three problems. Weights: #1 - 30%, #2 - 35%, #3 - 35%.
Closed book, closed notes. Be sure your answers are presented in a neat and wellorganized manner.
1. Answer True or False for each of the following statemen
Econ 600
Schroeter
Fall 2013
Exam #1
Do all three problems. Weights: #1 - 30%, #2 - 30%, #3 - 40%.
Closed book, closed notes. Be sure your answers are thoroughly explained and
presented in a neat and well-organized manner.
1. Answer True or False for each
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
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 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
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
Econ 600
Schroeter
Fall 2005
Exam #1 Solution Outline
1. a. False. Replace "if and only if" with "only if"
b. True.
c. True.
d. False. Replace "positive semi-definite" with "positive definite"
e. True.
2. U a convex subset of n . Then
cfw_
G : U convex (
Econ 600
Schroeter
Fall 2007
Exam #2 Solution Outline
1. a. True
b. False. Change "for all u , v n " to "for all u , v n such that F (u ) F (v ) ."
c. False. Change "strict global maximum" to "strict local maximum."
d. False. Change "closed" to "compact"
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 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 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 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 2012
Exam #2 Solution Outline
1. a. False. Change if and only if to only if.
c. False. Change
b. True.
*
dF *
(a1 ) = F x * (a1 ); a1 to dF (a1 ) = L x * (a1 ); * (a1 ); a1
da
a
da
a
(
)
(
)
where L() is the Lagrangian defined by
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 ,