Econ 600 Schroeter Practice Problem Set #2 1. f : is differentiable and has a local maximum at x * . In addition, f (k ) ( x *) = 0 for k = 1, 2, 3,K , n - 1; and f (n ) ( x *) 0. (In other words, all of the derivatives of f () , from the 1st through the
Econ 600 Schroeter Practice Problem Set #1
2 1. Let F : 2 be defined by F ( x1 , x 2 ) = x13 + 2 x12 x 2 + 2 x 2 . Let x* = (1, 2 ) and x = (1,1) . Solve for the value of implicitly defined by
F ( x * + x ) = F ( x *) +
2 F (x *) x + 1 x F (x * +x ) x. x
Econ 600 Schroeter Practice Problem Set #3 1. Consider the constrained maximization problem:
w.r .t . x1 , x2
max F ( x1 , x 2 ) subject to g ( x1 , x 2 ) = c,
where F () and g () are differentiable, real-valued functions, and c is a constant. We showed i
Econ 600
Schroeter
Fall 2013
Exam #2 Solution Outline
1. a. False. Change maximum to minimum.
b. True.
c. False. Change to .
d. False. Change if and only if to only if.
e. False. Change xn n 1 converges to B . to xn n 1 converges to inf xn n1 .
2. a. If x
Econ 600 Schroeter Practice Problem Set #4 1. Let F : n x m x1 be a differentiable function of a vector of arguments (x, z, ) , where x and z are vectors of choice variables of dimensions n 1 and m 1 , respectively, and is a scalar parameter. Consider the
Econ 600 Schroeter Practice Problem Set #6 1. There are two goods: 1 and 2. Both goods are available at two different kinds of stores: regular stores and bargain stores. Prices for goods 1 and 2 at regular stores are p1 and p 2 respectively; at bargain st
Econ 600 Schroeter Practice Problem Set #5 1. (Simon and Blume 21.28) Let F ( x; a ) : n x be convex in a for every x n and let X be a subset of n . Consider the problem:
max F ( x; a ) subject to x X .
w.r .t . x
Let Z (a ) be the set of global solutions
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 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 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 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 #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