answers are worth at most half-credit!
1. Let
C = {x 2 Rn | kxk1 ? 1}
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 ˆx be a point on the boundary of C. Explicitly
identify the supporting hyperplanes of C at ˆx. [CAUTION ! Di?erent 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 defined 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