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 experience that the Hessian conditions for concavity of the Cob
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 are homothetic if (x, y )I (x , y ) implies (tx, ty )I (
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 , x2 ) = = =t max cfw_tx1 , tx2 t max cfw_x1 , x2
1/2
m
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,
provide a counterexample. (In all answers where you provide
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 them. Please let me know if you nd mistakes or inadequa
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 in the
opposite direction is also considered an eigenvec
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 P such that xP y if and only if xRy and not yRx. a) Is
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? For what values of k is f a concave function? Explain yo
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 with a twocommodity model, where the two commodities a
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
point (x, f (x) lies entirely above the graph of f . You sh
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, 000. His shady
brother-in-law has given him inside in
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 by a utility function of the form
T
U ( x1 , . . . , x
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 may be used as inputs in production, some of which may
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
for some continuous function f .
Derive the consumers d
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 the numeraire with a price of 1 and let p be the price
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 example to show that if
P is not necessarily complete.
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 generation and marginal utility theory by about a century.1 In 1
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 for each good. His Indirect utility function. His Hicksia
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, of what degree? Explain your answer. If not, show that
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 , x1 + 2x2 ) . a) Draw some indierence curves for Mary. b)
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 worse than the indirect utility of the (p, m)
budget. T