Exercise 1.15

The budget set, B, is defined on page 20. In the book a compact set in Rn is defined as a set that is closed and bounded (p.426).

'Closed' means all the boundary points are in the set (p.422). I hope this is obvious for B, but if not try drawing a diagram. A more technical answer is that the sets X={x:x in Rn+} and Y={x:px<=y} are both closed so B=(X intersection Y) is closed by Theorem A1.4.4 (p.423).

'Bounded' means the set is enclosed in a closed ball (p.423). Note that p>>o means that q = min(p1,p2,...,pn)>0. Now take the closed ball with centre 0 and radius y/q. The budget set is enclosed in this ball. Draw a diagram again!

So we have shown compactness.

For convexity, take w and x in B and show that aw + (1-a)x is also in B where 0<=a<=1 (p.412).

Exercise 1.32

There is a hint for this exercise (p.523).

Define v*(p,y) = G(v(p,y)) where G is some arbitrary, positive monotonic transform. What you have to show is that x*(p,y)=x(p,y), that is the demand functions from v* and v are the same. Good luck!