Proving that a Cobb-Douglas function is concave
if the sum of exponents is no bigger than 1
Ted Bergstrom, Econ 210A, UCSB
If you tried this problem in your homework, you learned from painful experien
Answers to some old prelim questions: 1. Ms Blue and Mr. Green A Ms Blue has same preferences as Mr. Green, since her utility function is a monotone increasing function of his UB = eUG B Preferences a
Answers to Econ 210A Midterm, October 2010 Question 1. f (x1 , x2 ) = max cfw_x1 , x2 A. The function f is homogeneous of degree 1/2. To see this, note that for all t > 0 and all (x1 , x2 ) f (tx1 ,
Midterm Examination: Economics 210A
October 2011
The exam has 6 questions. Answer as many as you can. Good luck.
1) A) Must every quasi-concave function must be concave? If so, prove it. If not,
provi
Questions and Answers from Econ 210A Final: Fall 2008 I have gone to some trouble to explain the answers to all of these questions, because I think that there is much to be learned by working through
Eigenvalues
If the action of a matrix on a (nonzero) vector changes its magnitude but not its direction, then
the vector is called an eigenvector of that matrix. A vector which is "flipped" to point i
Name
Midterm Exam, Econ 210A, UCSB, Nov 5, 2007
Try to answer ve of the six questions, including Question 6. 1) Suppose that the preference relation R is complete and transitive. We dene the relation
Final Exam, Econ 210A, December, 2008 Please use a blue book for your answers. 1) Firm A has production function f (x1 , x2 ) = x1 where k > 0. a) For what values of k is f a quasi-concave function? F
Lecture Notes on Separable Preferences
Ted Bergstrom UCSB Econ 210A
When applied economists want to focus attention on a single commodity
or on one commodity group, they often nd it convenient to work
Rooftop Theorem for Concave functions
This theorem asserts that if f is a dierentiable concave function of a single
variable, then at any point x in the domain of f , the tangent line through the
poin
Homework Problems on Expected Utility
Fall 2009, Econ 210A UCSB
1. Randy Variable is an expected utility maximizer with a von Neumann Morgenstern utility function v (x) = x1/2 . He has a wealth of $99
Preliminary Examination in Microeconomics August 25, 2010 Question 1 Harry consumes just one commmodity and he will live for T periods. His current preferences over consumption streams are represented
Lecture Notes on Production and Cost Functions Ted Bergstrom UCSB Econ 210A
A very general model of production possibilities can be described as follows: Suppose that there are m goods, some of which
There are two commodities. Someone has an indirect utility function
v (p1 , p2 , y ) = G A(p1 , p2 ) +
y y 1
1
Assume that the price of good 2 is xed at p2 and that
p0
A(p1 , p2 ) =
f (, p2 , y )d
p1
Some Old Prelim Questions 1. Mr. Green consumes two goods, X and Y . His utility function is U (x, y ) = ln (x + 2y y2 ) 2
where x is his consumption of X and y is his consumption of Y . Let Good X be
Homework Problem 1
Let be a binary relation on X . Consider the relation P on X dened
so that xP y if and only if NOT y x.
A) If
is transitive and complete, show that P must be transitive.
B) Give an
Notes on Uncertainty and Expected Utility
Ted Bergstrom, UCSB Economics 210A
November 21, 2011
1
Introduction
Expected utility theory has a remarkably long history, predating Adam Smith
by a generatio
Midterm Exam, Econ 210A, Fall 2008 1) Elmer Kinks utility function is mincfw_x1 , 2x2 . Draw a few indierence curves for Elmer. Find each of the following for Elmer: His Marshallian demand function fo
Name Midterm Exam, Econ 210A, Fall 2010 Answer as many questions as you can. Put your answers on these sheets. Question 1. Let f (x1 , x2 ) = for all x1 0, x2 0. A) Is f a homogeneous function? If so,
Midterm Exam, Econ 210, Fall 2009 Answer Question 1 and any 3 of the other questions. Question 1. Mary Granola consumes only two goods and her utility function is 2 U (x1 , x2 ) = (mincfw_2x1 + x2 , x
Notes on Indirect Utility
How do we show that the indirect utility function is quasi-convex?
We want to show that if v (p, m) v (p , m ), then the indirect utility of
the convex combination budget is