Econ 600 Schroeter Fall 2006 Homework 1. A firm employs two inputs, x1 and x 2 , to produce one output y = f ( x1 , x2 ) , facing fixed prices for the output and inputs: p, w1 , and w2 . The firm's problem is
w.r .t . x1 , x2
max (x1 , x 2 ; p, w1 , w2 ),
Econ 600 Schroeter Fall 2008 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 2008 Homework (Note: These problems are very closely related to the problems in the required homework set from fall 2006 for which a solution outline is available on the web page.) 1. A firm uses two inputs, x1 and x 2 , to produce
Econ 600 Schroeter Fall 2006 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 well-organized manner. 1. Answer True or False for each of the following stateme
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 2007 Exam #1 Solution Outline 1. a. True. b. False. Change "negative semi-definite" to "negative definite." (Then could also add that x * is a "strict local maximum.") c. True. d. False. Change the inequality to "
2 d 2F* (a0 ) F x
Econ 600 Schroeter Fall 2008 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 well-organized manner. 1. Answer True or False for each of the following 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 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 #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
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
Econ600
Schroeter
Fall2007
Homework solutionoutline
1.a.Lagrangians forthetwoproblems:
L ( x l ) = F( x a ) - l g( x a )
a
i
(
L ( x l ,m ) = F( x a ) - l g( x a ) - m h x a )
a
ii
FONCsforsolutionto(i)with a =a 0 include:
L *
F *
g *
i
x (a0) 0l*(a 0 )
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 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 ,