Instructions: Liberal partial credit will be awarded IF you show your work. Unjustied
answers are worth at most half-credit!
1. Let
C = fx 2 Rn j kxk1 1g
which is the unit ball in `1(Rn).
(a) In the case that n = 2, draw a careful picture of the ball. Is it convex? Why or why
not?
(b) For a general dimension n, let ^x be a point on the boundary of C. Explicitly
identify the supporting hyperplanes of C at ^x. [CAUTION ! Dierent points on
the boundary may well have multiple supporting hyperplanes.]
2. Show that a set C Rn is convex if and only if its intersection with any line is convex.
3. Show that the epigraph of a continuous convex function f : Rn ??! R is a closed set.
4. In the notes on consumer behavior we have dened what it means for a preference relation
to be non-satiated. A preference relation is called locally non-satiated provided for every
> 0 there is a y 2 X with ky ?? xk < such that y x.
(a) Show that local non-satiation implies non-satiation.
(b) Typically, a consumer has limited resources and hence must meet a budget. If p is a
price vector and x is a typical bundle, then the price of that bundle is given by the
inner product hp; xi. Given a utility function, a rational consumer chooses a bundle
which is a solution to the constrained optimization problem
max
x2X
u(x)
subject to hp; xi m:
where m is the maximum buying power available to the given consumer.
Show that under local non-satiation a utility maximizing bundle x? must be such
that hp; x?i = m.
(c) Show that if is strictly convex, i.e., if x0 x then, for all 2 (0; 1) ; x+(1??)x0
x, then, for each price vector p > 0 there is a unique solution to the constrained
maximization problem.
DUE: My oce by 5 p.m. Friday, December 18 together with your signature to the statement below.
I certify that the work submitted for this nal examination is my own work and that I have
not consulted with any other person concerning the examination problems or their solution.
NAME:
SIGNATURE:
.

Instructions: Liberal partial credit will be awarded IF you show your work. Unjustied
answers are worth at most half-credit!
1. Let
C = fx 2 Rn j kxk1 1g
which is the unit ball in `1(Rn).
(a) In the case that n = 2, draw a careful picture of the ball. Is it convex? Why or why
not?
(b) For a general dimension n, let ^x be a point on the boundary of C. Explicitly
identify the supporting hyperplanes of C at ^x. [CAUTION ! Dierent points on
the boundary may well have multiple supporting hyperplanes.]
2. Show that a set C Rn is convex if and only if its intersection with any line is convex.
3. Show that the epigraph of a continuous convex function f : Rn ??! R is a closed set.
4. In the notes on consumer behavior we have dened what it means for a preference relation
to be non-satiated. A preference relation is called locally non-satiated provided for every
> 0 there is a y 2 X with ky ?? xk < such that y x.
(a) Show that local non-satiation implies non-satiation.
(b) Typically, a consumer has limited resources and hence must meet a budget. If p is a
price vector and x is a typical bundle, then the price of that bundle is given by the
inner product hp; xi. Given a utility function, a rational consumer chooses a bundle
which is a solution to the constrained optimization problem
max
x2X
u(x)
subject to hp; xi m:
where m is the maximum buying power available to the given consumer.
Show that under local non-satiation a utility maximizing bundle x? must be such
that hp; x?i = m.
(c) Show that if is strictly convex, i.e., if x0 x then, for all 2 (0; 1) ; x+(1??)x0
x, then, for each price vector p > 0 there is a unique solution to the constrained
maximization problem.
DUE: My oce by 5 p.m. Friday, December 18 together with your signature to the statement below.
I certify that the work submitted for this nal examination is my own work and that I have
not consulted with any other person concerning the examination problems or their solution.
NAME:
SIGNATURE:
.